Verification of the Lateral Drainage Component of the HELP (Hydrologic Evaluation of Landfill Performance) Model Using Physical Models

EPA/600/2-87/0-19 July 1987 VERIFICATION OF THE LATERAL DRAINAGE COMPONENT OF THE HELP MODEL USING PHYSICAL .MODELS by P. R. Schroeder and R, L, Peyton L'.S. Artsy Engineer Waterways Experiaent Station Vicksburg, MS 39180-0631 Interagency Agreement Number DV96930236-01-0 Project Officer D. C. Aaanon Landfill Pollution Control Division Hazardous Waste Engineering Research Laboratory Cincinnati, OH 4526a HAZARDOUS WASTE ENGINEERING RESEARCH I .MORATORY OFFICE OF RESEARCH AND DEVELOPMENT U.S. ENVIRONMENTAL PROTECTION AGENCY CINCINNATI, OH 45268 ------- NOTICE This document has been reviewed in accordance with U.S. Environmental Protection Agency policy and approved for publication. Mention of trade navies or corn-rercij 1 products does r.ot constitute endorse- ment or recommendation for use. ii ------- FOREWORD Today's rapidly developing and changing technologies and Indue. -Ill products and prectices frequently carry with then the Increased generation af solid and hazardous wastes. Th>:se materials, If improperly dealt with, car threaten both public halth an', the environment. Abandoned waste alt's anJ accidetital rsleases of toxic and hazardous substances to the environaint also have important environmental and public health implications. The Hazardous Waste Engineering Research Laboratory assists in providing an authoritative and defennible engineering basis for assessing and solving these proolemi. Its products support the policies, progrsas and regulations of the environ- mental Protection Agency, the permitting and other responsibilities of State and local governments and the needs of both large and small businesses in handling their wastes responsibly and economically. This report describes a study conducted to verify the lateral drainage component of the Hydrologic Evaluation of Landfill Performance (HEU'i comt.uter ¦odcl using laboratory drainage data from two large-scale physical Mdeli of landfill liner/drain aystees. Drainage teatr were run to ixislat toe ef:ects that drainage length, slope, hydraulic conductivity and depth of saturation have on the lateral drainage rate. The drainage results were compared with HELP model predictions and numerical solutions of the Boussinesq equation for unsteady, unconfirmed flow through porous media. Detailed water budge.s, test liumarlis, and tables of the test data collected are published in appendices ujder a separate report covi»r. Thomas R. Hauser» Director Hazardous Waste Engineering Researc.i Laboratory Hi ------- ABSTRACT Two large-scale physical models of land±ill liner/drain systems ver* consti-ucted to examine the effects that the length, slope and saturated hydraulic conductivity cf the drain layer, and the u«>	------- CONTENTS Foreword iii Abstract lv Figures . ..... vl Tables xi Acknowledgments xiii 1. Executive Summary 1 Purpose and scope 1 Results of drainage tests 2 Verification of the HELP model 4 Conclusions and recommendations ... 6 2. Introduction . ........ 7 Background ....... . 7 Description of HELP model ....... 9 Purpose ar.d scope 13 3. Model Design, Preparation, and Instrumentation . . 14 Model design and construction 14 Test preparation , 18 Instrumentation .......... 21 4. Experic«?nral Design and Procedures 27 Experimental design 27 Test procedure and data collection 30 Data reduction ....... 32 5. Physical Mcdtl kesults ... ........... 33 Saturated-depth profiles . 33 Drainage rate and saturated depth . 33 Drainable porority 52 ------- 7IGVRES Sur.be r Page 1 Plan view of experimental components of the physical aodels . 1 2 Cutavay s4 de view of physical xcdel 16 3 Cu:avay end view of phvsicai cedel at the trough: 17 4 Compact ion curves for buckshot clay sell 20 5 Grafn-size distributions of the sand drainage aedia 22 6 Sketch of piezoneter atid transducer installation 23 ? Saturated depth profiles for unsteady drainage during 24-hr ralr.fi.ll test at 2-percer.t slope and during 6-hr rainfall test at 10-percent slope using fine sand in short raodel 34 8 Saturated deK th profiles for unsteady drainage during 6-hr rainfall rest using fine sar.d In long model a; '.lopes cf 2 and 10 percent 35 9 Saturated deoth prof ilea for unsteady drainage during 6-hr rainfall test using coarse sand In short sodel at slopes of 2 and 10 percent . 36 10 Saturated dept! profiles for unsteady drainage during 6-hr rainfall test using coarse sand in Ion® model at slopes ;-£ J and 10 percent 37 11 Saturated depth profiles for steady-state drainage in both physical models 38 12 Head profiles for unsteady drainage during 24-hr rainfall test at 2-percent slope and during 6-hr rainfall test at 10-percent slope using fine sand ir. short xodel ..... 39 13 Head prcfiles for unsteady drainage during 6-hr rainfall test using fine sand in lon^ aodel at slopes of 2 ard iO percent 40 vi ------- Number Page 14 Head profiles for unsteady drainage during 6-hr rainfall test using r.o&rse sand In short model at slopes of 2 and 10 percent 41 15 Head profiles for unsteady drainage during 6-hr rainfall test using coarse sand In long acdel at slopes of 2 and 10 percent 42 16 Head profiles for steady-state drain.iga in both physical models 3 \7 Average saturated depth and drainage rate as a function of tiae for 24-hr rainfall test using fine sand and for 6-hr rainfall test using coarse sand in short model at 2-percent slope . 44 18 Average saturated depth and drainage rate as a fur.ction of tiae. for 6-hr rainfall test using both sands in short model at 5-percent slope 45 19 Average saturated depth and drainage rate as a function of tiae for 6-hr rainfall rest using both sjo43 in short model at 10-per-enc slope 46 20 Average saturated depth and drainage rate as a function of tiae for 6-hr rainfall test using both sands in long oodel at 2-percent slope 47 21 Average saturated depth ar.d drainage rate as a function of time fox 6-hr raiufall test using both sands in long model at 5-percent slope 48 22 Averse saturated depth and drainage rate as a function of tiae for 6-hr rainfall test using bsth sands in long model at 10-p«;rcent slope ..... 49 23 Unsteady drainage rate as a function of average saturated defth for fine srnd in both physical models ...... 50 24 Unsteady drainage rate as a function of average saturated depth for coarse sand in both physical models 51 25 Measured drainage rate followng rainfall vs. average head above the liner compared to HELP drainage equation predictions for fine sand in both physical models at slopes of 2 and 10 percent ....... 62 vli ------- Number Page 26 Measured drainage rate following rainfall vs. average head above the liner compared to HEL? drainage equation predictions for coarse sand in both physical models at slopes of 2 and 10 percent 63 17 Measured drainage rate and average saturated depth vs. time cospared to HELP prediction for fine sand in the shoit model at slopes of 2 and lt> percent ... . 65 28 Measured drainage rate and average saturated depth «s, ties compared to HELP prediction for fine sand in the long model at slopes of 2 and 10 per-ent 66 29 Measured drainage rate and average saturated depth vs. tiue compared to HELP prediction for coarse sand in the short model at slopes of 2 and 10 percent ......... 6? 30 Measured drainage rate and average saturated depth vs. tiee ccmpa^d to HELP prediction for coarse sand in the long model at slopes of 2 and 10 percent 68 31 Measured drainage rate and average saturated depth vs. tine cospared to HELP prediction for fine sand in the long model at 10-percent slope using mean values for hydraulic conductivity and dralnable porosity • 69 32 Measured drainage rate and average saturated depth vs. time compared to HEL? prediction for coarse sand in the «hort oodel at 10-percent slope using mean values for hydraulic conductivity and dr»lr>sble porosity 70 33 Predicted »s, actual drainage rate using measured drainage from short model to predict drainage from long model ...... 7j. 34 Predicted vs. actual drainage rats using measured drainage at 2-percent slope to predict drainage at 10-prcent slope ... 76 35 Predicted vs. actual drainage rate using measured drainage at y - 6 in. to predict drainage at y • 12 in. ........ 78 36 Predicted vs. actual drainage rates using measured drainage at al • 15.3 in. to predict drainage at aL - 30.5 in 80 37 Predicted vs. actual drainage rate using measured drainage from short model to predict draincge from long model having the same value of aL ...... ....... 82 B-l Comparisons of hydraulic conductivity values for steady-state drainage to values for unsteady drainage aa estimated by the HELP equation and the numerical Boussinesq solution .... 94 viii ------- Mumber Page 9-2 Comparisons of hydraulic conductivity estieates by the HFL? equation to est isa-tes by the numerical Boussir.esq solution for steadv-state and unsteady drainage 96 3-3 Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Boussinesq solution for unsteady drainage following rainfall and frota presaturated sands 97 B-4 Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Boussinesq solution for unsteady drainage froa fine and coarse sands ... 99 8-5 Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Boussinesq solution for unsteady drainage from short and lor.g physical models 100 B-6 Comparisons of hydraulic conductivity estimates by the HFL? equation to estiaates by the numerical Boussinesq solution for unsteady draicagt from models at 2-, 5- and 10-percent slopes . 101 B-7 Comparisons of hydraulic conductivity estimates by the KELP equation to estiaates by the numerical Boussir.esq solution for unsteady drainage ct depths of saturation ranging from 0 to 7 in., 8 to 14 in., and 15 to 20 in 102 B-8 Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Boussinesq solution for unsteady drainage from models at 2-percent slot^e with depths of saturation tanging trom 0 to 7 in., 8 to 14 in., and 15 co 20 in 2-9 Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Boussinesq solution for unsteady drainage from models at 10-percent slope with depths of saturation ranging from 0 to 7 in., 8 to 14 in., and 15 to 20 in. . . C-l Measured drain; rate vs. average head compared to numerical Boussinesq solutions for fine sand in both physical models at slopes of ? and 10 percent . C-2 Measured drainage rate vs. average head compared to numerical Boussinesq solutions for coarse sand in both physical models at slopes of 2 and 10 percent 108 C-3 Measured drainage rot* and average saturated depth vs. tine compared to numerical Boussinesq solutions for fine sand in short model at slopes of 2 and 10 percent 110 . . 103 . . 104 . . 107 lx ------- Number Page C-4 Measured dtair.age rate and average saturated depth vs. tias compared to numerical 3ouss"!nesq solutions for fine sand In long sodel at sio?«3 of 2 and 10 percent Ill C-5 Measured drainage rate and average saturated depth vs time compared to numerical Boussinesq solutions for coarse sar.d in short mode] at slopes of 2 and 10 per cent 112 0-6 Measured drainage rate and average saturated depth vs. tine compared to numerical Boussinesq solutions fcr coarse sand in Isng socle 1 at slopes of 2 and 10 percent 113 C-? Measured head profiler for unsteady drainage during 24-hr rainfall test at 2-percent slope and during 6-hr rainfall test at 10-percent slope compared to numerical Boussinesq solutions for fine sand ir. short model 114 C-8 Measured head profiles for unsteady drainage during 6-hr rainfall test compared to numerical Boussinesq solutions for fine sand in long model at slopes of 2 and 10 percent 115 C-9 Measured head profiles for unsteady drainage during 6-hr rainfall test rompared to numerical Boussinesq solutions for coarse sand In short model at slopes of 2 and 10 percent 116 C-IQ Measured head profiles for unsteady drainage during 6-hr rainfall test compared to numerical Bous-ir.esq solutions for coarse sand in l^ng model ct slopes of 2 and 10 percent 117 ------- T\BLES Nusber Page i ¦ JB» 1 Experimental Conditions £cr Unsteady Drainage Tests with Rainfall 28 2 Experimental Conditions for Steady-State Drainage Tests 29 3 Experieental Conditions for Drainage Tests Usir.g Presaturated Send 30 4 Drainable Porosity 53 5 Values for Drainable Porosity Constant, DPC 54 6 Drainage Uses 55 7 Coaparison of Cocnutea Hydraulic Conductivities for Steady-State Drainage ... 57 8 Coaparison of Computed Hydraulic Conductivities far Unsteady Drainage 59 9 Regression Coefficients for Hydraulic Conductivity as a Power Function of y 61 10 Effect of Length in the HELP Drainage Equation . 72 11 Regression Analyses Sunitary for Evaluation of the HELP Lateral Jrainage Equation . ..... ~4 12 Effect of Slope in the HELP Drainage Equation 75 13 Effect of Average Depth of Saturation in the HELP Drainage Equation .... 77 14 iffect of al in the HELP Drainage Equation 79 15 Effect of Length in the KELP Drainage Equation Given ronstant aL 81 A-l Hydraulic Conductivity Esticates for Steady-State Drainage .... 87 A-2 Hydraulic Conductivity estimates for Unsteady Drainage Following Rainfall ..... 86 xl ------- Nuaber Pagt A-3 Hydraulic Conductivity Estimates for Unsteady Drainage froo Presaturated Sand 91 8-1 Hydraulic Conductivity Regression Analysis Sunmary 93 B-2 Hydraulic Conductivity Regression Analysis for Unsteady " Drainage Following fi3infall , . . . 95 xil ------- ACKNOWLEDGMENTS Toe authors would like to express their sincere appreciation to Mr. Sidne; Sagsdale of the Environmental Engineering Division (EED), Environ- mental L?joratory (EL), U.S. Army Engineer Vaterva s Experiment Station (WES) and Kr. ?>oy Leach cf the Geotechnical Laboratory (GL), VES, for their help in designing, constructing, and operating the physical models. The authors wish to acknowledge Mr. Stafford Cooper, GL, for designing the jacks for the model; the Engineering and Construction Services Division, VES, for constructing and preparing the physical models fc* testing; and the Instrumentation Services Division, WES, in particular Kr. Thomas McEven, for instrumenting the physical ncdels. The authors also wish to acknowledge Mr. Thoias E. Schaefer, Jr., and Mr. Roy Wade of the EED, VES, for running the drainage tests. The authors would like to- thank Mr. Anthony Gibson and Mses. Charlotte Harness, Anita Zltta and Kathy Smart of the £ZD, WES, for their many contributions of tech- nical support leading to the completion of this study. In addition, the authors would like to acknowledge the support and general supervision of Mr. F, Douglas Shields and Dr. Raymond L, Montgomery of the EED, WES. The authors would also like to thank Dr. Robert Havis ar.d Dr. Bruce McEnroe of the EED, WES, for their techrical review and the Information Products Division, WIS, for preparation of the final figures and editorial review. xlii ------- SECTION 1 EXECUTIVE SUMMARY PURPOSE AND SCOPE This study was conducted to test and verify the liqaid management tech- nology for lateral subsurface drainage in covers and leachate collection systems. The specific objective was to verify the lateral drainage component of the Hydrologic Evaluation of Landfill Performance (HELP) Model (1,2) and other regulatory and technical guidance, provisions and procedures developed by the U.S. Environmental Protection Agency (JSEPA) (3), The HELP model is a computer model that generates water budgets for a landfill by performing a daily sequential simulation of water movement into, through and out of the landfill. The model produces ear'.mazes of depths of saturation and volumes of runoff, evapotranspiration, lateral drainage, and percolation. Lateral drainage is computed in the model as a function of the average depth of saturation above the liner, the 3lope of the surface of th# lir.ir, the length to the drainage collector, and t".e hydraulic conductivity of the lateral drainage layer (1). Therefore, to accomplish the objective of this study, the lateral drainage rate was measured as a function of the hydraulic conductivity, slope, lenrth and depth of saturation of the lateral drainage layer in large-seal physical models. The measured average depths of saturation, drainage rates and drainage times in the physical models were then compared with HELP model predictions ,md numerical solutions of ths Boussinesq equation, which applies Darcy's lav t > unsteady, unconfirmed flow through porous m»dici. Two large-scale physical models of landfill liner/drain systems were con- structed and filled with a 3-foot (iij sand drain l.yer overlying a 1-ft clay liner, A 2-inch (In.) layer of gravel was placed under tHe liner to collect seepage from the clay. The models were instrumented to measure the water table profile, subsurface lateral drainage rate, water implication, runoff and percolation through the liner. Evapotranspiration and other water losses were estimated from the water budget for each test. Ths models have adjustable slope, ranging from 2 to 10 percent in rhi3 study; and different lengths, one being 26.5 ft and the other being 53,5 ft. Several drainage tests were run on each configuration of the models by applying water as rainfall to the surface ot the send layer, and then measur- ing the water table along the length of the models and the lateral drainage rite as a function of time. Lateral drainage rates arid wa'et table profiles vere measured during periods of increasing, decreasing and steady-state drain- age rates. In these drainage tests, two drainage lengths were 1 ------- compared'—25.4 ft and 52.4 ft. Three slopes were examined—approximately 2, 5 arid^lO percent. Sands of two hydraulic conductivities we je used—4 x 10 cent Ir.eter.'second (era/sec) (fine sand) and 2.2 x 10 ca/sec (coarse sand) as measured In soil testing pemeaaetars. Four rainfall events were examined—a 1-hour (hr) rainfall at 0.50 inches/hour (in./hi), a 2-hr rainfall at 1.50 in./hr, a 6-hr rainfall at 0.50 in./hr and a 2i-hr rainfall at 0.125 lr,./hr. Also, water wcs applied to the sand for a long period of tLie (generally mors than 36 hr) at a rainfall intensity which would maintain the average depth of saturation In the sand at 12 in. In addition to thest drainage tests, the sand was saturated, predominantly from tha bottom up for several test conditions, and then allowed to drain. In total, mot« than sixty .* tests were performed. A complete block sxperimertal design was used to examine the effects of drainage length, slope, hydxaulic conductivity, depth cf saturation, rainfall intensity and rainfall duration on the lateral subsurface drainage rates. The block design was selected because it provided the aoct data with the least tine and expense for construction and model preparation. Several slopes and rainfall events caulc, be examined quickly since very little else was tequirad for changing these test conditions. Aiso, the time requirements and costs for running an additional test with a different slope or rainfall were less thin 10 percent of the requirements for preparing the model for a different sard. Additional rainfall events were examined in lieu of replicates since the lateral drainage rate as computed by the HELP model does not directly cor.slder the effscts of rainfall Intensity or duration. Also, since a complete block design was used, the effect of a change in a variable is directly examined under oultiple test conditions, reducing the need for replicates. RESULTS OF DRAINAGE TESTS A comparison of profile shapes for the depth of saturation alcn-£ the length of the drainage layer indicates significant differences between the rising saturated-depth profile (during filling) and the falling saturated- depth profile (during draining) for the same average depth of saturation (y). The prof litis are steeper naar the drain when filling than when draining. The difference Is greater for higher infiltration rates. Steady-state profiles are very similar to the profiles Cor draining. The drainage rate for a given average depth of saturation was greater during the filling portion of each experiment than during the draining •portion. ThL% is consistent with the saturated-depth profiles whith show steeper hydraulic gradients r.ear the drain for filling conditions. Plots of drainage rates as a function of average depth of saturation also show that draiazge continues aftsi.* y has essentially reached zero. This Is presuced to be drainage of capillary water, commonly called delayed yield. An estimate of this capillary water volume when y had just drained to 0 in. based on an analysis of the experimental data Is about 0.1 in. (cubic inches per square inch) for the fine sand and 0.3 is. for the coarse sand. The drainag: results indicated that the drainable porosity of the sands decreased with increasing depths of saturation above the clay liner. In addition, the drainable porosity at all depths was considerably smallar than 2 ------- the value estimated from soli moisture dat? and other soil properties collected on the sands. Low dralnable porosity vsiues were obtained In part due to the delayed yield and capillary effects that results from the high drainsge race. However, the the presence and vertical distribution of entrapped air appear to be primarily responsible for the low dralnable porosities and the change in dralnable porosity with height, although no measurements of entrapped air were collected. All parameters required to compute the drainage rate by the KELP equation except the hydraulic conductivity were measured for each drainage test. Due to variable air entrapment and differences in placement, compaction and preparation of the sand drainage media, the hydraulic conductivity measured In a permeaneter in the soils testing laboratory differed significantly from the actual test values calculated from data on drainage rates and depths of saturation from the physical models. As described in the documentation report for the HELP model (1), the lateral drainage equation was developed to approximate numerical solutions of the Bousslnesq equation fcr on«»- diaensional, unsteady, unconfined flow through porous media. Therefore, the actual hydraulic conductivity for che drainage tests was estimated by adjusting ita value while solving the Bousslnesq equation until the results matched the measured drainage r>ites and saturated depths. The hydraulic con- ductivity estimates are summarized In Appendix A. Determining the hydraulic conductivity in this taanrsr provided the best estimate obtainable for each test since the Bousslnesq solution is the commonly accepted representation of the accual drainage process. Comparisons were made for both steady-state drainage during rainfall and unsteady drainage following cessation of rainfall. The computed hydraulic conductivity values differed significantly from the treasured values. For steady-stace drainage frcr. the fins sand, t!-e average computed value was only 8 percent greater than the measured value, while for unsteady drainage the average computed value was about 150 percent greater. For steady-state and unsteady drainage from the coarse sand, the average computed value was respectively 92 and 84 percent less than the measured value. For both sands, the average confuted hydraulic conductivity for unsteady drainage was twice as large as the computed hydraulic con- ductivity for steady-srate drainage. In analyzing the computed hydraulic conductivity values, It was apparent that hydiaulic conductivity decreased with Increasing y. This is consistent with the earlier hypothesis that the volume of entrapped air increased with increasing distance above the clay liner. A larger volume of entrapped air decreases dralnable porosity ana cross-sectional flow-through area, thereby decreasing hydraulic conductivity. The computed hydraulic conductivity values varied considerably between tests on the same sand, even In the same model without disturbing the placement of the sand between tests. Considerable variability occurred between tests having exactly the same configuration of sand, slope, Icr.gth, and depth of saturation, where only the rainfall intensity and duration differed. This variance was examined using an unequal three-way analysis of 3 ------- variance (ANC-VA) test to determine whether the computed hydraulic conductivity •¦fas a function of another variable beside! average saturated depth. The test variables used in the ASCVAs included type of sand, average saturated depth, slope, drainage length, rainfall duration and rainfall intensity. Ho effects of rainfall duratio-i and intensity could b<- discerned by inspection; therefore, the initial AKOVAs were run using depth, slope and length as the variables for data sets containing hydraulic cor.duc.ivity •stiaates for one type of sand. These ANOVAs indicated that the computed hydraulic conductivity estimates for both sands varied as a function of average saturated depth and slope. Additional A>'GVAs indicated that drainage length, rainfall intensity and duration did not significantly contribute to thc variance in the computed hydraulic conductivity values. No physical reasons are apparent for the variability of the hydraulic conductivity as a function of slope. Therefore, the variability due to slope probably arises from inaccuracies in the manner in which th# effects of slope are modeled by the, boussinesq equation. VERIFICATION OF THE HELP MODEL The drainage rates computed by the. HELP t.odel was compared with the results of the drainage tests in several manners. The hydraulic conductivity that was needed to yield the measurer* drainage rate for the sa=r. drainage length, slope, and aves.nge saturated depth existing in the drainage test was computed for several times during each ti st. Thi 3 hydraulic conductivity valje we9 cosrpa.?d with the vaJue measured in th* soils testing laboratory and the "alue estimated using tha 3oi.asir.esq equation. If the HELP equation accurately predicted the result? of the drainage ;ests, the hydraulic con- ductivity value would agree witn the measured rr estimated hydraulic con- ductivity value for the sand. If the h/drauliw conductivity value was greater than t'ae value obtained for the sar.d, the HEL"1 equation underpredicted the lateral drainage rate, Anorher method of comparison was to exanine the effects of changing a single variable on the lateral drainage rate measured in the drainage tests and predicted by the HELI* equation. The effects of drainage length, slope, average depth of saturation, and the head contributed by the slope of the liner were compared in this manner. The hydraulic conductivity of the sand at various depths of saturation was estimated for eacn test using thi Boussinesq solution of Davcy's law for unsteady, unconfined flow thrcugh parous media and the HELP latiral drainage equation. These hydraulic conductivity values were coopered to deteraine the agreement between the KELP model and the Boussinesq solution, for steady- state drainage, tne HELP model istiaates of th* hydraulic conductivity were 44 percer.t greater than the Boui sine?4 solution estimates. This result me&r.s that the HELP model underestine ed the steady-state lateral drainage rate predicted by the Boussinesq solut .on by 30 percsnt. The HELP estimates were 31 percent greater than the laboratory measurements for the f'.ne wind and 88 percent less than the laboratory measurements for the coarse s tnd. For unsteady drainage, the HELP model estimates were only 13 percent greater than the Boussinesq solution estimates which would underpmdict the lateral dnirage rate by 11 percent, Th« closeness of the estimates was not unexpected since the HELP lateral drainage equation was developed from numerical 4 ------- solutions of che Boussinesq equation for saturated unconflned l.-.teral flow through poreus media under unsteady drainage conditions. The t.nderprediet ion of the cumulative lateral drainage volume would be expeced te be very snail since the removal rate of wai:er frosc the drain layer by ail other means is much smaller than the lateral drairage rate. Consequently, che effect of dif- ferences in tha predicted and actual drainage times are sral!. The differences between t'.se laboratory measurement of the hydraulic con- ductivity and either of the two estimates computed from drainage data were much larger Chan the differences between the estimates. The HELP model and Boussinesq equation predicted very similar drainage rates at 2-percent slope but the HELP model predicted lower drainage rates at 10-percent slope. Unlike the laboratory measurements, the hydraulic conductivity in the drainage tests .arled as a function of the depth of saturation apparently due to entrapment of air in the sand. This phenomenon makes it very difficult to model the lateral drainage process and produce good agreement betwe-so the predicted and actual results for drainage rate ar.d depth of saturation as a fui.ction of t late • An analysis was perforaed to deternine how well the lateral drainage equation in the HELP model accounts for the effects of drainage length, slope of the liner, average depth of saturation and head above the driin contributed by the liner in the estimation of the drainage rate. The drainage equation overestimates the decrease in drainage rate resulting from an Increase in length given the tame sand, slope, depth of saturation and head from the liner. Using the dringe rate for a drainage length of 25.4 ft to predict the rate for a length of 52.4 ft, the HELP model underpredicte3 che rate by 18 percent. The HELP equation overestimated the increase in drainage rate by 30 percent that resulted from an Increase in slope from 2 percent to 10 per- cent. Similarly, the HELP equation overestimated the increase in drainage rate by 2Q percent that resulted from increasing the height (head) of the crest of the liner from 15.3 te 30,5 in. above the drain. The effects of changes in the average saturated depth on the drainage rate ptedicted by the HELP equation agreed very well with the actual results. Since the HELP lateral drainage equation was developed t> approximate uua-.wriCPi solutions of the one-dimensional Boussinesq equation for unsteady, unconflned, saturated flow through -porous media, it cannot be expected to perform any better than the Boussinesq equation. Therefore, ic was necessary to compare the Boussinesq solutions to the laboratory measurements in order to form a basis for judging the significance of the differences between che HELP equation predictions ard the laboratory measurements, and between the HELP equation and the Boussinesq solution. To summarize, the Bouaslnesq solution after calibration still produced results significantly different frc* those measured in the drainage tests. The results obtained with che HELP sodel were generally as good or better than the Boussinesq solution. The HELP equition performed better on teats con- ducted with 2-pnrcent slope end the Boussinesq solution performed somewhat better on teats conducted at 10-percent slope. The dlfferen:ea between predictions by the two methods for a given set of conditions were small In comparison to the range of actual results. Siallarly, the differences between 5 ------- the predictions and the actual results were much larger than the differences between the HELp equation and the Bou«sInesq equation. CONCLUSION'S AND RECOMMENDATIONS The following conclusions and reccaaendations ars 3iade. Lateral drair.aps in landfill lin<.-r/drain systems is quite variable, probably due to air entrap- ment. The hydraulic conductivity measurement cade in the laboratory is qu'te different than rhe ir.-place value. Consequently, the estieacior. of the l?t- determine whether the effects ;.re unique to this ''ata set. Additional data should be collected for longer drainage lengths and greater slopes and from actu il landfill/liner syster.s. 6 ------- SECTION 2 INTRODUCTION BACKGROUND Landfills have coat to be a widely employed means for disposal of munici- pal, industrial and hazardous solid wasres. Storage of any waste m/.terlal In a landfill poses several potential problems. Among these is the p-jsslble con- tamination of ground and surface waters by th» migration of water or leachate from the landfill to adjacent areas. Given this potential problea, it is essential th3t the liquids oanageaent lechnclogy perform as expected over the life of the landfill. It is also essential that the performance of the tech- nology can be simulated or modeled with sufficient accuracy to design land- fills to prevent aigra:ior. of liquids froc the facility. The modeling of the moisture movement through landfills also provides important information for review of landfill designs and evaluation of the adequacy of the design and the limitations of the liquids management technology. This study was conducted to test and verify the liquid management tech- nology for lateral subsurface drainage in covers and leachate collection sys- tems. The specific objective was to verify the lateral drainage component of the Hydrologic Ev?luatior. of Landfill Performance (HELP) Model (1,2) and other regulatory and technical guidance, provisions and procedures developed by the U.S. Environmental Protection Agency (L'SEPA) (3). The USEPA regulatory provisions for lateral drainage layers require only that the depth of leachate buildup at the bottom oi the landfill should not exceed I ft and the construction materials for i>.aerate collection should be resistant to chemical attack and the physical forces exerted on them (3). USEPA technical guidance states that the drainage layer should be constructed to be at least 12 In. thick at a minimus slope <^f - percent %nd have a hydraulic conductivity of not less than 1 x iO cm/sic. Also, the drainage pipe system should be of appropriate size and spacing -o efficiently remove the leachate. It is believed that 4-in,-diameter pipes spaced 50 tc 200 ft apart would be adequate for removing leachate (3). The drainage layer in the cover shculi have the same specifications as above except that pipe drainage systems are not necessary, although free drainage must be provided at the perimeter of the cover (3). Subsurface drainage has been a subject of interest for at least 200 years as san attempted to drain marshes and reclaim land for health and agricultural purposes. The literature is filled with citations describing drawdown of a water table under steady-state conditions with pipes placed la parallel at a constant elevation. In the majority of these studies the impervious barrier 7 ------- soil layer was well below the elevation of the drain pipes. Drainage unier these conditions has been fully described and can be predicted vithin 10-percent accuracy. The phreatic surface is elliptical as lon& as the drains reaain unsubmerged and do not restrict drainage. Drilnage from soils where the drainage pipes are plactd on the surface of the barrier soil layer is less well defined, in general, the drainage for this condition is considerably slower, and the accuracy of the drainage rate estimate by Dupuit's law (Darcy's law for unccnfined flow) is slightly worse, though reasonably good. The drainage rate Is smaller because the cross- sectional area through which the water flows toward the drain is sraaller. The drainage rate is slightly overestimated because the flow is more curvilinear, violating the parallel flow assui.ption. Studies on subsurface drainage from soils above a sloped Impervious layer are not widely found in the literature. Preliminary findings on drainage through pipes placed well above the barrier soil layer have been repo-ed, but none of these studies examined drainage from the surface of the barrier soil layer as performed in landfills and in this study. The literature does not present equations to predict the drainage rate or the phreatic surface. The majority of the drainage equations reported in tha literature were developed using, 3 steady-state assumption of ? uniform recharge rate equal to the drainage rete. The drainage equation used in the HELP model assumed steady state to develop the basic form of a steady-state equation but was cor- rected to agree with the results of a numerical isodel for unsteady drainage. Drainage in the cover of a landfill or from the leachate collection system of an open landfill is clearly transient, and the phreatic surface profile may differ significantly from the elliptical profile obtained under steady-state conditions. The profiles will vary while the drainage layer fills and drains. Consequently, as the profiles vary, the drainage rates vary for -hi sane aver- age depth of saturation. The prediction of drainage rates is complicated by many factors. At low heads, unsaturated flow controlled by capillary action, soil moisture gradients and gravity can sSynificantly contribute to drainage. Matrix effects between sell layers can affect the drainage between the layers. Air can be entrapped in the l«>ers altering the head, hydraulic conductivity and phreatic surface profile. The unsaturated hydraulic conductivity of soil at a given moistur* content varies depending on whether the soil is wetting or drying. Field measurements of hydraulic conductivity, porosity, and field capacity are difficult to perform precisely and accurately. Soils are gen-rally not uniform and homogeneous. This study examines trar.«lent or unsteady drainage from the entire drain- age layer above a sloped barrier soil layer to verify the equaric- used in rSe HELP model* This equation was developed by extending equations oevclop»>* in the literature for simpler cases. This study provided much-needed information in three areas where the literature is lacking: transient dm. ainage, drainage from the surface of a barrier soil layer, and drainage from soils above sloped barrier soil layers. This study measured drainage rates .is a function of 8 ------- phreatic surface profile for two sands, three slopes, a range of saturated depths, and two drain spaclr.gs. DESCRIPTION OF KELP MODEL The HELP node I Is a computer model chat gererates »ater budgets for a landfill by performing a daily sequential simulation of water movement into, through and out of the landfill. The model estimates of depths of saturation and voluaes of runoff, evapotranspiracion, lateral drainage, and percolation. Laceral drainage is computed in the model as a function of the average depth of saturation above the liner, the slope of the surface of the liner, the length to the drainage collector, and the hydraulic conductivity of the lateral drainage layer (I): 0.16 2 (0.51 + 0.C0205 a L) K y ' " u . IU -\| 5 "3L] L2 (1) where Q - average lateral drainage rate for the time period, inches/day (in./day) a - slope, diaensiooless L - drainage length, inches K • hydraulic conductivity, in./day >• - average depth of S3turacicn, inches As presented in the HELP documentation report (I), this equation was u«.veloped from the Boussinesq equation for unsteady, unconfined laminar flow through porous media. The Boussinesq equation is obtained by combining Darcy's law with the Dupuit-Forchheiaer assumptions (also known as Dupuit's law and Dupuit's approximation) with the continuity equation. Dupuit's law is a steady-state equation for lateral flow: q - -X y (dh/dx) (2) where q « flow rate per unit width y ¦ depth of saturation above the impervious bed h • height of free surface above the drain x • horizontal distance from the drain 9 ------- This law assuae9 that the flow throughout the dep>.h of saturated soil is hori- zontal, Thus, the equipotent ial lines are vertical and the streaalinas are horizontal. The second assumption of this law is that the hydraulic gradient is nqual to the slop# of the :ree surface and does not vary with depth. The continuity equation for lateral drainage from porous aedia say be written as fallows f Csy/31> - -(jq.'ix) + R (2) wher: f • trainable porosity C ¦ Ciae R ¦ rate of vertical infiltration or evaporation into or out of the saturated soil Combining Equations 2 and 3. Che Boussinesq equation is obtained: f (37/aO - 3(K y (3h/ax))/3x + R (4) This equation applies Dupuit's law for unsteady conditions. Equations 2, 3> and 4 were developed for systeas with horizontal or mildly sloping impervious beds. Landfills typically have liners that are sloped from 2 to 30 percent, whih violates the funs of Equations 2, 3 and 4. If the flow is assumed to be parallel to the constantly sloped, impervious bed, then the equipotential lines would be perpendicular to the bed. Under these assumptions Equations 2, 3 and 4 can bi modified as follows q - -K y cos g (dh/dl) (5) f (jy/jt) - -{sq/al) + R cos e (&) f(jy/3t) - a(K y cos e (3h/ai))/ai + R cos a (7) where 0 " slope of the impervious bed h • y + I sin b 1 • distance froo drain along the bed slope Equation 7 is the form of the Boussinesq equation used to develop the HELP lateral drainage equation and the equation ®olvtd numerically in this study. 10 ------- To develop the HFL? equation, steady state was assumed (5y/3t - 0). This lap Lies that the phreatic surface profile does not differ significantly from the profile during unsteady conditions, particularly during periods when the ohreatic surface is rallir.g, Under steady-state conditions the infiltration rate (X) equals the overall average one-diaens ional lateral drainage rate 'Q; ; therefore. Equation 7 becomes Q - K d[y (dh/dl)/dl (jj'i This equation is nonlinear since both y and h are functions of 1. The bound- ary conditions for this equation are '.i • 0 at 1 ¦ 0 (9a) and dh/dl - 0 at 1 - L (9b) where L ¦ length of bed from drain to crest. Equation 8 vas linearized by setting dh/dl equal to the total change in head over the total length divided by the total length: dh/dl - (y + L sin S)/L ',10) o where y^ » depth of saturation at 1 - L. Equation 8 becosres Q - K (yo + L sin 3)(dy/dl)/L (11) Siaildrly, dy/dl was set to equal the average depth of saturation divided by half of the drainage length: dy/dl - y/CL/2) (12) Therefore, Equation 11 becoaes C. - 2 K v (v + I sin i)/J (13) o Equation 13 was then fs.-iver.eJ to replace y with a function of y since yo is unknown in the HELr ecdel. It was also corrected to agree with numeri- cal solutions of Equation 8 foe perioos when the phreacic surface is falling after the profile had reached steady state. The correction was xade for satu- rated depths racing froa 0 to 30 in,, for slopes ranging free : to 10 per- cent, and for drainage lengths ranging from 25 to 200 ft. The result of this correction yielded Equation 1 after replacing sir. 9 with a. The slope, a, in H ------- diner.sionless form is equivalent to tar. 8, which for snail slopes is approximately equal to sin 9. The spatially averaged depth of saturation is not constant in a landfill with respect to tine, and depending en the rainfall intensity, collection svs- teo design, and permeability of the soil lavers, can vary greatly in several hours. Since the drainage rate is a function of the average depth cf satura- tion, the drainage rate can also vary- greatly in several hours. As lateral drainage occurs from a drain layer without infiltration, the average depth of saturation continuously decreases; the drainage rate does likewise. Therefore, the HELP model solves Equation 1 as a function of time by applying it for a time step, yielding the average drainage rate for the tine seep. The tine steps were 6 hr for lateral drainage from above the top barrier soil layer of the landfill and 24 hr for the lower two barrier soil layers. The model reports the drainage rate from above each barrier coil layer daily, To obtain the daily value for the tcp barrier soil layer, the model averages the computed values from the four time steps. The units for drainage are inches/day (volume/day/surface area). The average drainage rate is a function of the average depth of satura- tion, which is a function of the average drainage rate. Therefore, the model solves for drainage iteratlvely by assuming the drainage rate, solving for the average depth of saturation and then solving for the average drainage rate. If the calculated drainage rate differs significantly^from the assumed value, a new estimate of the drainage rate is produced and the process is repeated until the estiva red and conputed values agree within 0.2 percent. The average depth of saturation is computed by dividing the drair.able water in the lowermost unsaturated segment in a subprofile by the drair.able porosity of that segment, and then adding this value to the sua of the thick- nesses of all saturated segments between that segment ar.d the barrier soil layer. In actuality, the average depth of saturation in a landfill, ph/sieal model or a two- or three-dimensional model would be determined by inf-sreting the depth of saturation over the area or along a path to the drain and divid- ing by the area or the length of the path. This is the method that was used in analyzing data from the physical models in ar. attempt to verify the drain- age equation. While the model neglects the lateral variation in saturated depth, the drainage equation used in the HELP model ins corrected to approximate the numerical solution of the one-diaensicnal Boussinesq equation as draining occurred. The numerically generated profiles were used to compute the average depths of saturation which were subsequently used to develop Equation i. Therefore, use of Equation 1 by the HELP model implies that the profiles are the same as those generated by the numerical solutions of the Boussinesq equation. 12 ------- PURPOSE AND SCOPE The obecclve of this study was to verify the lateral drainage component of the HELP model and other USEPA regulatory and technical guidance for leachate collection systems by determining the lateral drainage rate as a function of the hydraulic conductivity, slope, length and depth of saturation of the lateral drainage layer in large-scale physical sodels. the measured average depths of saturation, drainage rates and drainage times in the physical models were compared with the results predicted by the HELP equations for vertical and lateral drainage. Two large-scale physical models of landfill liner/drain systems were con- structed and filled with a 3-ft sand drain layer overlying a I-ft clay liner. A 2-in. layer of gravel was placed under the liner to collect seepage from the clay. The models were instrumented to measure the water table profile, sub- surface lateral c'rainage rate, water application, runoff and percolation through the liner. The models have adjustable slope and two drainage lengths, 25.5 ft and 52.5 ft. Drainage tests were conducted on both models at three slopes—2, 5 and 10 percent. Drainage from two dlfferrnt sands was studied in each model at each siope. Several drainage tests ware run on each configuration of the models by applying water as rainfall to the surface of the sand layer and then measuring the water table along the length of the models and the drainage rate as a function of time. The water was applied at several intensities and for several durations. 13 ------- SECTION 3 MODEL DESIGN, PREPARATION, AND INSTR'JMENT.YTION MODEL DESIGN" AND CONSTRUCTION Two physical models were designed and constructed to perform the labora- tory verification tests. The models, shown in Figures 1 and 2, were con- structed at two different lengths to permit the examination of the effects of length on drainage. One model was built to have a usable uepth of 5.0 ft, an inside width of 5.3 ft, and a drainage length of 25,4 ft. The other model had the same depth and width but had a drainage length of 52.4 ft. loth models were constructed of identical materials. The base of the models was constructed of steel soil test cars used fnr mobility studies. These cars are reinforced 1/4-in. steel tanks that are 21 ft long, 5.3 ft wide and 2.5 ft deep. The sides of the cars were extended upward 4 ft with marine grade, 1/2-in. plywood supported by 2-in. angle iron. Silicone sealant was used to fill the joints and cracks and to seal and prevent leaks. The models were placed at h 2-percent slope and fitted with supports for attaching 5-ft screw jacks at the upper end. A jacking structure was built to attach to either rodel which could raise the end of a fully loaded rrodel. The jack could increase the slope of the long model to 11 percent and the short model to 20 percent. The slopes of the models during testing were determined by surveying. Three troughs, approximately 12 in. deep and extending across the width of the models, were placed inside the lower end of each model as shown in Figure 3. The bottom trough was used to collect seepage through the clay liner. The middle trough collected subsurface lateral drainage at the bottou of the sand drainage layer. The top trough collected runoff from the surface of the sand. The bottom two troughs were constructed of 1/4-in. steel plates welded to the steel soil test car. The top trough was fabricated of galvanized sheet metal and bolt•d in place to the plywood walls. The sheet metal was scaled to the walls with silicone sealant. The runoff trough was drained to a 5-ft-deep, 32-in,-diameter (dia) sump tank via 1.5-dia polyvinyl chloride CPVC) pipe. The water level in the tank was measured with a steel measuring tape before and after a simulated rain event to determine the runoff volume. The drainage trough was also drained to a 5-ft-deep, 32-in.-dia susp tank via 1.5-in.~dia PVC pipe. The drainage sump tanks were automatically pumped out when filled to a height of about 44 in., requiring only about 5 minutes (min) to lower the water depth to about 10 in. The minimum time required to fill the tank was about 4 hr. The seepage trough 14 ------- RllAMf/t OHAtN Kill « Htll( AtHtN WAfl* IfVt HtCOfilifK 1 Alt>H i OHAl* I AUK tirvti, 0tA.il I 0«4'.V4u( •y 7"v4< £/lut tMAHwuct HtAOIHoS MC.UtAtO* "4 C.i'1 AfOO • iMrrMOl VAI VI Figure L. Plan vlev of experimental components of the physical models. ------- ft AIM CAUT OftlVe TRAIN structure RAIN CART NOZZLE PIIIOHITIN SAND CLAY FIITCN FABRIC XiMf(jm«4kl! MfcMJllAMe FILL MATERIAL XXio* * wr r™A •" 'vvv¦'¦ ^ ^ ^^-".'lCt>0 (T tippiho »UC«t jkMD OB"' FiR"re Cutaway eiAe » „jral roodt^• lew of P^0 ------- Rain Cart Witr Sub d ty Ho« • 3p'y Noiz1• Top of Stnd Lay* #.3' Runoll Tfou;h SAND S t • • I Cart V Top of City Ly«r GRAVEL Percolation Trough Or sin Pic. t Figure 3. Cutaway end view of physical model at the troughs. 17 ------- was periodically drained into a 4-15ter (i) graduated cylinder to measure the seepage volume. The seepage was drained from the trough via an attached 2-in.-dia rubber hose chat was sealed hecw.'sr, measurements. Both models were fitted with a device t~ spray water uniformly across the length of the surface of the sand drain layer. This device consisted of a l.5-in,-dia ?VC pipe aounted on a 12-ft-long aluminum cart that tracked back and forth on rails that tan the length o- the model. A nozzle that sprayed a uniform line of water across the width of the model was attached on each end cf the pipe. Each nozzle sprayed on one-half of the node! as the cart coved its maxir.ua distance is one direction. The height of the nozzles and the travel limits of the cart were adjustable to ensure that the entire surface of the sar.d wa3 sprayed. The water application rate or rainfall intensity was adjusted by changing the size cf thi nozzles or changing the water pressure applied on the nozzles. The intensities of flow rates could be varied between 0.02 and 6 in./hr or 0.06 and 10 galJons/ainute (gpa). The intensity during a test was determined by meaeuring the volume of water applied as a function of tiae. A stopwatch was used to measure the time ar.d a Badger Meter, Inc., Model Reeordall 12 water aeter measured the water volume. The nozzles usee in thw tests were Floodjet 1/8 K.25 316SS through 1/4 K1B 316SS that were manu- factured by Spraying Systems Co. Schematics of the models are shown in Figures 1, 2 and 3. Figure 1 shows a plan view of the experimental setup, including the layout of the two models, water and drain lines, instrumentation, and sump tan>s. Figure 2 shows a cutaway side view of one model including the troughs, drain lines, sumps, jacking structure and rain cart. Figure 3 shows an cutaway end view of the model illustrating the placement of the trough and the shape of the model, TEST PREPARATION After construction was completed, the models wr>re filled with three func- tional layers and a layer of fill dirt as shown in Figure 2. The top layer was a 3-ft sand layer for lateral drainage. The second layer was a 1—ft com- pacted clay liner designed tc minimize percolation. Ths third layer was a 2-in. layer of pea gravel to transmit the seepage from the clay liner to a drain. The bottom layer was a lC-in. layer of fill dirt to build up the pea gravel layer to promote drai-^ge. Fill material 10 in. deep was compacted in the bottom of the steel test car to fill the trapezoidal section of the model and to reduce the volume cf pea gravel required. The fill material was covered with a 30-ail, butyl rubber membrane, T-16 manufactured by Firestone. The meubrane was glued to the walls of the steel car to prevent percolation into the fill material. The glue was 0~580, a synthetic rubber resin dispersed in solvent and manufactured by Pittsburgh Paint & Glass. A 2-in. layer of washed creek pea gravel was placed on the impermeable membrane and in the seepage collection trough. This layer was designed tc have a high permeability and low storage potential in order to transmit the snail volume of seepage rapidly to the collection trough. The layer was 18 ------- covered with Bidia Type C3«. filter fabric manufactured by Monsanto to prevent migration of fines from the clay liner into the pea gravel, thereby protecting the ability of the pea gravel to transmit seepage, 7h ------- 1 1 5 110 - 3 10S u •%, ia n 100 cr Q 90 t 35 1 3.0 LEGEND 6 COMPACTION EFFORT, CE 12 O COMPACTION EFFORT, CE 21 O COMPACTION EFFORT , CE 58 9 AS COMPACTED IN MODELS 15.5 ta.O 20.5 23.0 25.5 28.0 WATER CONSENT (% dry wt) 30.5 Figure 4. Compaction curves for buckshot clay soil. 20 ------- Two different sands were used ia Che tests, One sand was Reid-Bedford sand with Windham clay nonuniferxly dispersed throughout. The clay content was about 10 percent ir the short model and about 13 percent In the long xodel. The dry density, as placed, was 106 pcf. This is a relative density of 96 percent, indicating very good compaction. The specific gravity was 2.7, and the coefficient of permeability measured in the sails laboratory was about 3.5 x 10" cn/sec. The porosity was C. 3?, the aaxirr.ua drainable porosity was between 0.21 to 0.25, and the wilting point was about 0.07. The grain-size distribution, of the s ir.d with the clay is shown in Figure 5. The ether sane was a local, ungraded washed creek sand. The dry density, as placed, was 107 pcf in the long model and HO pcf in the short model, indi- cating both were hi-hly compacted. The specific gravity of the sand was 2.7, and the coefficient cf permeability measured in the soils laboratory was 2.2 x 10 cm/sec. The porosit; wss 0.35 In the short model and 0.36 in the long sodsl. The maximum drainaole porosity was between 0.24 to 0,28 and the uilting poiv.t was about 0.C4 to 0.05. The grain-size distribution is shewn in t igure 5. INSTRUMENTATION Description and Installatin The lateral drainage rate from the sand layer ves determined by two devices, one for low flow rates and the other for high flow rates. Low flrw rater (below 0,08 gpta or C . 05 in./hr for the short r.odel and 0.02 in./hr for the long acdel) were determined by measuring the drainage volume as a function of time using a Weathertronics Model 60'. 0 tipping bucket rain gage. The tip- ping bucket collected thu water from th« end of the ?VC pipa draining t'.e drainage trough and discnarged the water into the drainage suap tank. A bypass was placed in th« PVC pipe to divert che drainage directly intc the suop tank during periods of very high flow races. Higr.er flow ratei {greater than 0.08 gpm) were determine.! fcv measuring the water level In the drairage suop tank as a function of time using a Vea:her Measure Corporation Model F553-A waci.r level recorder powered at 12 volts by a ZLCO Mo-.el 1060S low ripple batte'-y eliminator and charger. Pcie-pres'iure transducers were placed in the clay liner vim the trans- ducer exposed upward toward the sand layer. The face of the transducer was set slightly b'.low :he euirounding cop surface of che clay, T/j install a transducer, a hole about 5 to 6 in, deep and 6 in, in diameter was dug in the clay. The transducer and wire lead were aced in the hole jr.'; the removed clay was tamped back in place. The transducer was about 2.'^ in. long and 1 in. in dia-neter. The wire lead frcr Che transducer vas '.la:ed just below che surface of the cla/ to prevent interference with the 1attral drainage £roo the sand Ia%er. The trandccer installation is shown in Figure 6. Six CTtoaducers were placed in the long model ar.d five ware placed in the short xodel. Consolidation Electrodynamics Corporation Tyre 4-312 pressure pickups were fitted in a brass body with a porous stone sc the pressure trans- ducer couU only measure pore pressure, which in thio case was the head of 21 ------- U.I. SltNOAftO SUVC 'irrnlNO IN INCmCI u.e. iiano.md iicvc nuntcpt 70 HTOR'jnCIf R BIO 19 70 JO 40 tO 70 100 14000 50 w 70 E l 30 (00 too 60 10 i i GRAIN SIZE OS 0.1 N MILLIMETERS d.os o.ot o.oos o.oni COBBLES GRfiVEL SAND SILl OR CLflT \| coani. I rmr csni>sr 1 ncoiun 1 nxc LL \| PL IPI 106 NRI H.X I.P.CEUP A Keid-Hcdford S.ind Willi 102 Vlndluni Clay O Ungraded Washed Creek Snnd CLRS3IFICRII0N O SAND (SP), 11K1H/N: WITH GRAVEL 6SII.T SAND (Sl'-SH), BROWN ORflDHTI ON CURVE Figure 5. Grain-size dlfltrlhutlonn of the sand drainage media. ------- Transducer Lead 1/2* PVC Pipe Piezometer SAND 4* Lon v j HoIIqki / P o f o j S )i- S t o r. e Transducer Figur# 6. Sktcch of pliz'jtr-star and trmsdurrr InstaJ lacion. 23 ------- water on the transducer. The locations of the transducers wre app.-oxinately 3, 6, 11, 16 and 23 ft from the lip of the drainage trough in che short model and 3, 6, 11 , 23, 35 ar.d 50 ft from the trough in the long model, as shown in Figure 2. The transducers wr.re placed cn Che centerline dividing the width of the models. The elevations and locations of the transducers were determined by surveying. Piezometers were installed in the sand layer to provide a backup method for measuring the depth of saturation above the clay liner. These measure- ments were also used to calibrate the pressure transducers and tc check the accuracy of the transducers throughout the testing. Piezometers were con- structed by Attaching 1.5 in.-dii, 4-in. long-cylindrical porous stones to I/2 — ir> ,-dia ?VC pipes of approxinately 4-ft lengths. The piezometers were installed with the; stone end placed flush with the tcp of the clay liner and with the pipe extending straight up through the sand layer. The depth of sat- uration was determined by treasuring the depth to the water surface from the end of the pipe and then subtracting that v*i„e from the total le	------- bucket. The number of tips was counted and compared with the number r-scorded on the water level recorder. The drainage rates computed froir the tipping bucket data and the water level recorder data were compared at the oid/ang* of the flow rates and agreed well with each other. Seepage was collected in the xcdel for a week or longer anJ then drained into a 5-gal bucket or a graduated cylinder. The volume cf seepage was mea- sured with a graduated cylinder. The depth of saturation above the clay liner was treasured in two ways— manually using piezometers and automatically using pore pressure transducers. The piezometers were calibrated by aeasuring the ler.gch cf each piezometer and surveying the location and elevation of each piezometer with respect to the lips of the runoff and drainage troughs ana the clay liner. Tl.e end of each piezometer vaj placed flush with the surface of the clay liner. Prior to installation, the transducers were calibrated in the instrumen- tation laboratory to determine the linear response factors for a unit increase in head and the readings fir zero head. The responses from the transducers were recorded using a Digi'.rend Med el 210 digital recorder with a Hewlett Packard Model 6113A 5-volt EC power suopiy, a Doric Model 214 digital clock and a Doric Model DS-100 integrating microvoltceter. A Consolidated Electro- dynamics Corporation Type 8-108 DC bridge balance was used to convert the millivolt signals from the transducers to pressure readings in pounds per square inch for the experiments with the Reid-Bedford sand. This same equip- ment was used both in the instrumentation laboratory and in the rodel tests. After Installation, the locations and elevations of the transducers were surveyed with respect both to the l^s of the runoff ard drainage troughs ind the clay liner. After the sand was placed above the transducers, the trans- ducers were calibrated again in place to establish the readings which corre- sponded tr> iero head of water on the clay liner since the transducers were not placed flush with the surface of the liner. The transducers were also checked to determine if the linear response factors for a unit increase in head were the same rs in the instrumentation laboratory. The zeros and response factors were determined in place by saturating the sand layer and flooding the surface to establish free water suffices of known elevations r.bove the sand. Using the surveying data and the linear response factors, the zeros were calculated. The linear response factor.: were checked by computing the changes in the readings for several known changes in water level and comparing them with the factors determined in the instrumentation laboratory. Vhe response factors ag.eed well with the instrumentation laboratory data but the zeros varied considerably. The piezometers were read periodically throughout the testing to provide a backup for the transducers. The readings of the piezometers were compared with those of the transducers to better define the zeros for the transducers. After testing was completed, corrections to the zeros were determined from the comparisons. The corrections ranged from about -2 to 2 ir.. (±5 percent of maximum), though half of the transducer readings required little or no correc- tion. The zero of one transducer drifted with time in a constant manner, and the correction was made js a function oi time. After correcting the readings 25 ------- of che transducers which had piezometers associated with then, the readings of the other transducers were corrected. These readings were corrected by plotting the readings to form a profile of the saturation at about fi\e dif- ferent average heads for each test, A smooth curve was drawn through the pre- viously corrected readings and zero at the trough. The offset of the other readings for each pijt was noted to dcterair.e whether the offset was constant at all average heads and for each test. A constant offset at all heads ar.d for each test indicated that the zero was off ar.d required correction equal to the offset. If che offsets changed continuously, particularly in one direc- tion, it would indicate that the readings drifted. Readings of only one transducer drifted and the transducer was located at a piezometer. If the offsets increased or decreased ur.iionnly with increasing average head, it would indicate that the linear response factor was incorrect. Hone of the linear response factors required correction. Nonsysteaatic changes in the offsets indicated the variance in the transducer readings. The standard devi- ations of che readings were abouc 0.25 in. 26 ------- sect:jk - EXPERIMENTAL DESIGN AXT? PROCEDURES EXPERIMENTAL DESIGN A coirpiete block experimental design was used tc examine the efiects of drainage length, slope, hydraulic conductivity, depth of saturation, rainfall intensity and rainfall duration on the lateral subsurface drainage rates. The block design was selected because it provide;," the nost data with the least time and expense for construe tier, and ncdel preparation. Only two nodels cculd be built due to their costs, and only tve sands could be exacinec due fo the unavailability of another of significantly different permeability. Several slopes and rainfall events could be examined quickly since very little tine was required for changing these test conditions. Also, the time requirements and costs for running ar, additional test with a different slope or rainfall were less than LQ percent of the requirements for preparing the r»odel for a different san-j, Additional rainfall events were examined in lieu of replicates since the lateral drainage rate ss computed by Equation 1 does not directly consider the effects cf rainfall intensity cr duration. Also, sir.ee a complete block design was used, the effect of a change in a variable is directly exaoir.ed under multiple test conditions, reducing the need for replicates. A sunr.ary of the rcdel tests is presented in Tables I, 2 and 3. Two drainage lengths were compared—25.4 ft and 52.i ft. Three slopes were examined—approximately 2, 5 a^d 10 percent. Sards of two hydraulic con- ductivities were used—4 x 10 cc/sec (fine sand) and 2.2 x 10 c./sec Ccoarse sand). Four rainfall events were examined—a 1-hr rainfall at 0,50 in./hr, a 2-hr rainfill at 1.50 in. ,'hr, a 6-hr rainfall at 0.50 in./hr and a 24-hr rainfall at 0.125 in./hr. Also, water vae applied to the sand for a long period of time (generally more than 36 hr) sc a rainfall intensity that would maintain the average depth of saturation in the sar.d at 12 in. In addition to these drainage tests, the sand vas sacurs'ed, predominantly from the bot tea up for several test conditions, and then allowed to drain. The order of the testing was arranged to accocr.odate xanpover and equip- ment restrictions. Ail testL were performed on a given sand before replacing cne sand since it would have been very costly and tice-consuming to continually change the sand. Also, the placement of the sand would not have beei' identical each time, which would have been a source of error in the comparisons of the other variables such as drainage length and slope. Ail rain events were performed for a given test condition before changing the slope of the model. This was done to minimize manpower requirements ard 27 ------- TA3LE 1. EXPERIMENTAL CONDITIONS FOR UNSTEADY DRAINAGE TESTS WITH RAINTALL Model Painfull Rainfall Rainfall Test Type of Length Slcpe Intensity Duration Volume Setut) Sand (f c) (") ' (In./hr) (hr) (gal) 1A Fine 25.45 1.8 o.sa 1.0 48.5 1» ft It II 1.69 2.0 284,2 IC It ft It 0.56 6.0 284.4 ID If f* It 0.14 24.1 287,0 IE ft It It 1.69 2.0 282.2 2A 11 II 4.9 0.58 6.0 291.6 3A It II 10.4 0.56 1.0 48.6 3B i» If It 1.74 2.0 291.7 3C it It If 0.57 6.0 291.1 2D it 11 M 0.14 24.0 291.0 4A it 52.45 2.1 0.57 1.0 98.5 4B it It It 1.70 2.0 588.0 4C it It tf 0.54 6.3 593.0 ',D ft tl 11 0,14 23.5 587.0 Ik ft It 5.0 0.57 6.0 591.0 6A f» • t 10.3 O.S7 1.0 98.6 6B ft ft H 1.70 2.0 589.2 6C ft 1 II 0.54 6,2 581.0 6D ft ft II 0.14 24.6 592.2 ?A Coarse* 25.45 1.9 0.58 1.0 48.6 7B It II 1.73 2.0 291.6 7C »» If II C, 56 6.0 291.6 7l> ft It It 0.14 27.9 338.7 71 It It II 0.14 24.2 290.9 8a 1* It 5.4 1.73 2.0 291.4 BB it It It 0.59 5.9 291.1 9a II It 10.0 0.58 1.0 48.6 9s t» 1* ft 1.73 2.0 291.0 9C It It It 0.58 6.0 291.6 SD It If 11 0.12 29.6 290.9 9E ft It If 0.14 24,0 291.0 (Contin' je^) 28 ------- TABLE 1, (CONCLUDED) Model Rainfall Rainfall Rainfall Test Type of Length SI -)oe Intensity Duration "o;uce So cup Sand (ft) (%> (in./hr) (hr) (sal* I OA Coarse** 52.45 2,3 0.57 1.0 1GB 11 629.4 II 1.70 2.0 539.2 IOC (I II 0.56 6.1 591.0 10D ft It II 0.14 24.0 591.0 11A tl It 5.2 1.62 2.1 589.4 i IB II f) tl 0.57 6.0 590.4 12A 1 II 9.9 0.57 1.0 98,6 12B If H H 1.71 2.0 589.2 12C tl II tl 0.57 6.2 581.0 12D ft I It 0.14 24.0 590.4 12E ~1 It 1! 1.70 2.0 589.4 * Hydraulic conductivity ¦ 3 x 10 ^ cm /sec ** Hydraulic conduct ivity - 2 .2 x JO"1 cm/sec TABLE 2. EXPERIMENTAL CONDITIONS FOR STEABY-S TATE DRAINoGE TESTS Avg. Model Depth of Rainfall Drainage Test Type of Length Slope Saturation Rate Rate Setup Sand (ft) (") (in.) (in./hr) (in./hr) 1 fine* 25.45 1.8 12.6 0.0446 0,0336 2 if II 10.4 13.2 0.0714 0.0543 3 fl 52.45 10.3 12.7 0.0589 0.0249 4 Coarse** 25.45 1.9 U .4 0.0928 0.0708 5 It »» 10.0 11.3 0.2070 0.1688 6 If 52.45 2.3 11.3 0.0464 0.0230 7 It M 9.9 11.8 0.1212 0.1050 _3 * Hydraulic conduct:'-"ley - 3 x 10 em/sec ** Hydraulic conductivity ¦ 2.2 x 10 cm/sec 29 ------- TA3LE 3. EXPERIMENTAL CONDITIONS FOR DRAINAGE TESTS USING PRESATITRATED SAND Test Sertrs r>-p» of Sand Model Length (in.) Slope Initial Avg, Depth of Saturation (in.) 1 2 3 4 5 6 7 8 9 10 U F Lne* Coarse** 305.4 629.4 305.4 629.4 8 I LQ 20.95 16.79 27.35 24.74 9.24 29.79 19.14 2.44 27.75 21.18 14.62 * Hydraulic conductivity ** Hydraulic conductivity 3 x 10 \|tn/sec 2.2 x 10 cm/sec because It uculd have been difficult to achieve exactly the same slope again. The order in which the slopes were tested was not constant to accomodate manpower and equipment scheduling. The scheduling introduced sone randomness into the order of testing the effects cf slope. Similarly, the order of the rainfall events for different test conditions was varied to accommodate the work schedules for personnel. This scheduling introduced seme randomness into the order of testing the effects of differences in rain events. TEST PROCEDURE AND DATA COLLECTION Each test was run in one ot three manners. Lateral drainage was measured from a presaturated sand; from a sand before, during, and after a simulated rainfall event; or from a sand while attempting to maintain a constant aver.age depth of saturation of 12 ia. The test procedures for the three manners vere very similar; only the procedure for applying the water to the model varied. The procedures for running a test vete as follows; 30 ------- 1. Prepare model for testing. This included preparing the sand by plac- ing or leveling; preparing the rainfall cart for desired rainfall intensity by changing nozzles, nozzle heights and end stops; changing the slope of the model; draining sump tanks; and preparing the recorders. 2. Initialize test. This included starting the recorders, setting the time, draining the lateral drainage trough, recording tect conditions, measur- ing initial depth of water in runoff sum? tank, and recording the initial water meter reading, 3. App]y vater to the model as prescribed for the test. This Included adjusting the water application rate and the spray pattern. Note peculiarities cf the test including unusual environmental condi- tions, problems, and variations from the test procedure. 5, After the rainfall was completed and runoff ceases, measure the final depth of water in runoff sump tcnk. 6. Measure water levels in piezometers periodically. Drain tanks as required. 8. Continue testing until the heads become so low as to nearly expose a transducer to air and then stop drainage or start a new test. 9. Collect data from recorders and reduce the data. Prior to calibration of the pore pressure transducers, water was added slowly to both ends of the model. Ir. this manner the rand was saturated pri- marily from the bottom up, which forced rtie air out oi the sand. Following the calibration, the sand was drained and the drair.^ge rate and heads were measured continuously while ^raining. Drainsge from presaturated sand was measured for only four test ;onditiotis since saturalicr. was required only for calibration. For the tests in whi-h a constant average head of 12 in. was aaintained, water was applied to the model to increase -he head rapidly to 12 in. This water was applied at a rate of about 0.5 ir,,/hr. After reaching a head of 12 in., the rainfall intensity wa3 decreased to the intensity estimated as necessary to maintain the head. The desired intensity was determined by adding the evaporation rate, the rat# of leakage and seepage, and the drainage rate which corresponded to an average head of 12 in. during a previous rain- fall for the same physical test conditions, '."he intensity wa« adjusted during the test if the head did not r«""-i.n constant near 12 in. The rain was con- tinued for at least 12 hr aftai the hr.ad remained nearly constant. The evaporation rate ranged from about 0.2 to 0.3 in,/Jay while raining, as estimated by the differences in the water application rate and the lateral drainage rate during the steady-state drainage tests. Th« rate was dependent or. building temperature and on the spray nozzle. The building temperature 31 ------- ranged from aboot 58F co 9QF, but for most tests the temperature ranged between 70°F and 82®F, Leakage varied directly as a function of head. Both models leaked, but the short model was more watertight than the long model. The leakage rates for the short model were measured to be about 0,02 in./day at a head of 20 in. and about 0.01 in./day at a head of 12 in., while the leakage rates for the long model were measured to fce about 9.12 in./day at a head of 20 in. ar.d about 0.0? in./day at a head of 1-' ir., Neither model leaked at heads below 6 in. since all of the seams in the model were above this depth. The long model leaked more because It suffered greater deflections upon loading due to its greater length. This caused more seams to rupture and leak. These leakage rates we-e used to correct the water budgets for each test. The majority of the drainage tests encompassed measurements before, dur- ing, and after a specified rainfall event. In these tests, a specified quan- tity of water, either 48.5 or 291 gal for the short model and either 98.5 or 591 gal for the long model, was sprayed across the entire surface of the sand in a specified period of time—1, 2, 6 or 24 hr. These quantities of water correspond to about 0,57 in. and 3.4 In. of rain, respectively, for both models. During these tests, the head increased from a starting point of about . 4 in. to a peak value occurring about 30 sin after the rainfall ceased, and then returned to about 4 in. when the test was stopped. Several tests were allowed to continue until the head was nearly zero inches. DATA REDUCTION The readings of the pressure transducers were recorded on a digital recorder at an interval ranging from 10 to 50 tain, The reading for each transducer was converted into head of water at the location of each trans- ducer. The reading was converted by subtracting the zer<-> for the transducer and then multiplying this value by the linear response factor for the trans- ducer to convert the millivolt value to pressure head in inches of water. The head was then corrected for changes in the barometric pressure and temperature using the readings from a transducer installed to monitor these changes. The heads were then corrected for the differences between the elevations of the transducers and the elevations to the surrounding clay liner. After the actual heads were computed, the heads throughout each model at each time period were averaged using the trapezoidal rule. The head at the lip of the drainage trough was assumed to be zero, and at the upper end the head was assumed to be the same as at the uppermost transducer. The volume of drainage was r- corded continuously on a strip chart. The drainage rate was determined by reading the change in volume of drainage collected in the sump tat for an interval of time or the volume of drainage, as tip3, passing throw.h the tipping bucket during an interval of time. These volumes were then divided by the interval of time to yield the average drainage rate. This rate was recorded as the drainage rate at the center of the time interval. The intervals ranged from 15 rain to 3 hr depending on how rapidly the drainage rate was changing, l.ie drainage rate was then correlated with the heads by matching the time of the readings. 32 ------- SECTION 5 PHYSICAL MODEL RESULTS This section presents the results of the drainage tests. These results include measurements of the saturated depth as a function, cf distance from drain, measurements of saturated depth and drainage rate as a function of time and, similarly, measurements of drainage rate as a function of saturated depth. Explanations of these results are presented in the next section alofg with comparisons between these results and the HELP model predictions. SATURATED-DEPTH PROFILES The depth of saturation was measured using pressure transducers at various distances from the drain. Plots of depth of saturation versus distance f rotn the drain are shown in Figures 7-10 for various times during the experiments. The depth measurements at a given time were used to compute the average saturated depth (y) for that tiae by the trapezoidal rule. The values of y are noted in the figures for all saturated-depth profiles shown, a comparison of profile shapes indicates significant differences between the rising saturated-depth profile (fill conditions) and the falling saturated-depth profile (drain conditions) for the same y. The profiles are steeper near the drain when filling than whea draining. Th«? difference is greater fcr higher infiltration rates. Profile plots are shown in Figure 1.4 for steady-state discharge at slopes of 2 and 10 percent (a. • 0.02 and 0.10) when y equaled approximately 12 in. Steady-state profiles are very similar to the profiles for draining. Similar plots for the hydraulic head above the drain profiles are shove in Figures 12-16. All profiles shown in Figures 7-16 weris t^ken from the 6-hr rainfall experiments, except for the fine sand profiles in the short model at 2-percent alope, which were taken from the 24-hour rainfall experiment. DRAINAGE'RATE AND SATURATED DEPTH The variations in average saturated depth and drainage rate with time art shown in Figures 17-22. Again, these figures represent 6-hr rainfall experi- ments with the exception noted above. In some cases, the entire thickness of the drainage layer approached saturation prior to the end of rainfall. As a result, the curve of average depth versus time for a few cases approaches a plateau near the point of maximum average ------- 9 35 3a « »y-14 sa riLL a—Oy-2S.84* DRAIN 9-14.78 * DRAIN ~—•5- 9.86* DRAIN 2* SLOPE "O 25 20 IS ia 5 10 IS 2P 2% DISTANCE TROW DRAIN * (FEET) #—y»<2.32 fill a—05.21,88' drain ~—« crem Figure ?. Saturated depth profiles for unsteady drainage during 24-hr rainfall teat at 2-percent slope and during 6-hr rainfall teat at 10-percent slope using fine sand in short model. 34 ------- s i T y R A t £ D D £ P T H 2* SLOPE a—Ov-2S.9r Of:AIN ~ y-tS.13" DRAIN •—¥- 7 23 yRAIN 23 -I zs 3a DISTANCE FRCf DRAIN trerrj 10* SLOPE y-fS.» FILL DRAIN DRAIN DRAIN 2! .?! * IS.S2 a.M* a ~ 19 I IS T 25 29 25 30 DISTANCE fR0N DRAIN I 35 I 0 crem —r~ 45 1 sa S.'i Figure 8. Saturated depth profiles for unsteady drainage during 6-hr rainfall test using fine sand in lcng oodel at slopes tf 2 and 10 percent. 35 ------- 30 2* SLOPE fill DRAIN CRAZH 2,33* DRAIN zs 20 IS 10 s a 15 6 19 a DIS~ANC£ FTON WU1N * iFECf) s A t U R A T £ D 0 £ P T M I N c M r 2fl IS 10 -I y- 4,it" FILL -"v-ta.ao* DRAIN " V- ~ a?* DRAIN -y~ a.SB DRAIN 10% SLOPE Figure 9. DISTANCE rC« DAAIM » CFtTTi Saturated depth profiles for un»ttady drainage during 6-hr rainfall east using coarse sand in short aodel at slopes of 2 ar.d 10 percent. 36 ------- 2 SLOPE —>-13. 9' rju. a-"0 9.18,S! • OTAZN 13,34' DRAIN •—•- .88 CRAXN >S 29 2S sa s 3£ 46 ss e distance fwcm au» * cram 0% SLOPS tt—a 6. 78* r.LL 3—88* ORAlN v- S.S8* DRAIN •—V- 7 ?«• 5RAIN ®—1 .98 DRAIN ¦Ok. 3£. SS e 15 44 9 DISTANCE IHKCN WAIN * C«ETJ Saturated depth profiles for unsteady drainage during 6-hr rainfall test using coarse sand ,n long model at alopea of 2 and 10 percent. 3? ------- s A T U "» T E D 0 £ P T M I N C M E S 29 16 10 SHORT MOOEL .314. r3¦«. i n.nxe o-a.82t.COARSE •»«.ass.coarse -T_ is is i 39 —r~ 2S r- 38 —r- 3S T l ¦4S m OISTamcC PRlSl MUXN * crtTT> Figure 1t. S«irurat«d depth profiles for steady-state drainage in both physical models. 38 ------- "3S ?S S 0 3 S 9 OISANC£ FRON DRAIN , :FEE~2 —nu. 9<1 " SPAIN »4.3S* DRAIN •—•N-37.I3* DRAIN 13* SLOPE DISTANCE FROM DRAIN * CFCm :igjre 12, Head profiles for unsteady drainage during 24-hr rain- fall test ac 2-percent slope and during 6-hr rainfall test at 10-percent slope using fine jaid in short model. 39 ------- —"5-21 .78* FILL a—o H-.32,22 * ORAJN "~ZI . 74 5RAIN -13.90 CRAIN 2% S 2S 39 3S a distance fson draim » tfrrrj —•h-47.73* FILL B—0^53.95' DfUlN ~—»fi-4?,78* DRAIN »R-4l .82* PRAlN 10% 3 IS 28 ZS 3a 3S DISTANCE nw» DRAIN > CFECTi Figure 13. Head profilts for unsteady drainage during test using fine sand in long aodtl at slope 10 p«rc«nt. 40 ------- * •h-10.67* rlLL a—e^is.-M* DRaIN DRAIN DRAIN 2S .a-' 29 S 0 S 2* SLOPE a i is 25 29 a 19 DISTANCE FROM DRAIN * CFEET3 ?s 39 2S 28 IS jt' 10* SLOPE s 9 2S te 28 S 19 a distance from drain * crcrrj cigure 14. Head profiles for unsteady drainage during 6-hr rainfall teat using coarse sand In short node.', at slopes of 2 and 10 percent. 41 ------- -to r 35 J a.8-4" fill 3—BZ.2C.7S' DRAZN ~ h-23.se* ORAIN •—•R-12.13' DRAIN .-a 20 -i 2% SLOPE n 1 1 1 1 r 10 IS Zli 2S 30 3S 40 6 S3 SS DISTANCE FROW DRAIN CTEST2 N E A D A 8 n v e D R A I N ea 78 i «—>y3fl,97* rUX B—OFr-4S.68* DRAIN •—*-.<0.68 DRAIN —h-38.3«* DRAIN •—•N-Sa.SS* DRAIN 10* SLOPE DISTANCE FROM DRAIN * ------- —<»0 ai8,«ThC a—ca-0.104, fine ~ a-0 . 85 S, COARSE 0 '00.COARSE .M- 3S jr' 30 25 S a SHORT MOOEL s a 5 •S a 18 DISTANCE FROM DRAIN CFEET5 ias, fine a—« o-0 .023. CCARSE ~—* a9 . as« , COARSE LONG MODEL SS distance from drain » crcrr> Figure 16, Head profiles for steady-state drainage in both physical models. 43 ------- 80 2i FINE SAND AVtRAQC BICTM 8RAINAOI NATS 20 Hi z o s - 30 z k. w a HI 20 C 10 < t; •d > « .03 10 .00 10 20 40 0 TIM I PROW START OF RAIN (HOURS) HI z w z ~» ui a o * as- 20- 1»- 10- -o.s COARSE SANO Figure 17. • AVIRAM 6t»TN O ORAINAOf RATI tL a < s ui o m a ¦0,1 "5.0 ¦ I M i I I < \| • I n • • I I • > . ¦ J<1 ------- so-j ' FINE SAND • AVKAA4B BSPfH 0 ORAiMAOl RATI TIME PKOM STAHT OP RAIN 'HOUS> l-o .ae -4.1 $ Ml o.as -0.00 21- 10- X u X 1. M a m 9 < c Ml » m COARSE SAND • tvciuaa oiPfH B DRAMAS RATI -0.# -0.4 30 30 Tim FROM START OF RAIN (HOURS! Figure 18. Average saturated depth and drainage rate as a function of tine for 6-hr rainfall test using both sanda in short model at 5-p«rcent slope. 45 ------- so- -4 «UAU oarm PINE SAND SRAIKAat RATI 40 30: 10 20 40 10 0 TIMC MOM START Of KM (N0U9> a AVtlAOt IIM* COARSE 3AN0 10- OKAIHAdt »4T1 m m X 0 1 1« 1 o 10 kl 0 < 1 kJ 10 0 20 40 90 ae TIMC MOM STAl'T OP (AID (HOURS) Figure 19. Average saturated depth and drainage race as a furct: tine for 6-hr rainfall test using both sands in shor< aodel at 10-percent slope. 46 ------- SO AVINAM 9IPTH BKAIHA&g RATI 40 30 m 30 10 I^IIIIHIIt\|IHIIMII\|l 10 «0 30 40 ?0 BO 80 7mi pnom start or rain fwouiits s u « X o « c w > mhMm • 4VKRA4I QtPTN COARSE SAND iao 140 TIMS MO $TMT OP RAIN t»OUX> Figure 20. Average saturated depth and drainage rate as a function o tiae for 6-hr rainfall test using both sands in long node at 2-percent alop«, 47 ------- .Ill I FINE SAiO ' • AVIRAOI 6IPTM Q t)««IMAOI »ATI Ml 11 I 40 T» >0 •0 TO '^.2 «T to TIME MOM JTi'«T Of RA»K (HOWiSS » AVIMA9I 6«»tw o »niMai Miiti COARSE SANiJ TIME PROM START OF RAIN (N0UR9) Figure 21. Average saturated depth and drainage rate as a funetio clae for 6-hr rainfall test uslrg both sands in long a at 5-parcent slope. 48 ------- ,23 to .««¦•<«! depth fine ;and (Ktmidl S«T1 20 30 0.18 m 0.10 10- 0,00 Baa^pMHPaaaaa,iBHVv 10 0 20 30 40 TIME FROM snm OP RAIN '.HOURS) 28 20 Z o z 18- X k. s "»• « K m T-r- 0 COARSE 3AN0 • vi«a« etPTM a dramas* »ati 10 "~r <¦ 20 30 40 Tine FROM START OP RAIN (HO'JRS) -0.4 ¦0.4 Figure 22, Average saturated depth and drainage rate as a function of time for 6-hr rainfall test using bot* sands in long model at 10-percent slope. 49 ------- a 175- SHORT MODEL. 3 Q £ a ' sa- « 125- £ J e e i ®e- u • A7S- < * * 3 Z a as»- J. j < ¦< X 4 o a 925- 1 a aa#- ' a >5 20 25 AVERAQE DEPTH (Inch®#) 3a 3 O •>» o c UJ H < OS < DC Q a ao- a as- LONQ MODii a a? a ae- • as- a a-t-3 a 13 a az- a ai- a aa-v. s i a 6 ?s aa as AVERAGE DEPTH (lnch««) pigur« 23. Unsteady drainage rate aa a function of average saturated depth for fine sand in both physical aodels. 50 ------- fchORT MODE ? a i s » Q( x • I O a 83- a a e ia e AVESACE D£?TH (lnch««) i i J8 a «zs< a ase- LOMG MOOGi. a ; 5 j « » '®«i e u B Z- a »?s- ui < ec z < It a 82S- Q 3 a a-aa-L a z « e a ia * ia e ia 22 AVifl,v:iE DE?TH (Inehaa) Figure 24, Unsteady drainage race- ?.? a function of average saturated depth for coara-i sand in both physical aodels. 51 ------- coarse sand. Similar hysteresis occurrid ^ith ths fine sand. The da-.a for Che filling cycle were not plotted becausa they crossed over the Iir.es for higher slopi. For the same y, the dnimga rate during filling is greater than during draining. This is consistent with the profiles in Figures 7-16, which show steeper hydraulic gradients near t.ie -irain for filling conditions. The curves also show that drainage continues after y has essentially re?-hed zero. This is presumed to ae drainage of capil'.ary water, commonly called delayed yield. As estimate of this capillary water volume when 7 had just drained to 0 in. based on an analysis of the experimental data is about 0.1 in. (cubic inches per square inch) for the fine sad and C.3 in. for the coarse sand. DRAINA2LE POROSITY The y and drainage r-te data presented in Figures 17-22 were used co cco- pute drainable porosity at various heights within the tiodel. The volume of drainage water collected while the average saturated djpth fell from y, to y,, was divided by the total /olusa contained in the model between y, af.d y,. TRis cumber was used as an estimate of the drainable porosity within the region between y, and y?. The results are shown in Table 4. It is apparent that the drainable porosities in Table 4 (ranging from 0.01 to 0,20) are less than those cited in Section 3 (ranging from 0.21 to 0.28). Tha latter velues were naiHtnmi values based on soil moisture and porosity measurements taken after all experiments were concluded. The values in Table 4, particularly for high j values, are low due to delayed yield and capillary effects resulting from the high drainage rate. Nevertheless, significant reductions in drain- able porosity appeared to exist with increasing height above the clay liner. A closer examination of these data using regression analysis resulted in the following predictive equation for drainable porosity (IjP) at a point located a vertical distance y (in inches) above the clay liner: DP - DPC (0.955 - 0,0327 y) (14) where DPC » drainable porosity constant. Values for DPC are iisxed in Table 5 for eight experimental cases. Based on an examination of the data, a lower limit for DP was set for this study at 0.02?i DPC. It is assumed that the presence and vertical distribution of entrapped air were primarily responsible for the change in drainable porosity with height, although no measurements of entrapped air were obtained. Table 6 shows the time for y to fall from one level to another. As expected, the time for y to fall from 18 to 12 in. was significantly less than from 12 to 6 in. because of the larger discharge rates at larger y values. However, the smaller drainable porosities at latger y values also contributed to the shorter drain times. For the coarse sand experiments! the times for y to fall from 12 to 2 ia. ranged from 0.5 to 1.4 days for the 25,4-ft-iong model and from 1.1 to 4.0 days for the 52.4-ft-long modal. The fine sand experiments experienced much lor ,er drain times and therefore were not monitored long enough for y to reach 2 in. In all experiments, y drained from 18 to 12 in. in less than about 1 day. 52 ------- TABLE U. DRAINAB1.E POROSITY Uralnable Porosity Between y. and y„ Type of Slope Sand (Z) Fine Fine Fine Fine Fine Fine Coarse Coarse Coarse Coarse Coarse Coarse 2 5 10 2 5 10 2 5 10 2 5 10 Drain Length (ft) y, - 30 y2-25 y, - 25 v„ - 20 y j - 20 y„ - 15 y, - i> y2 - 25.A 25.A 25. A 52.A 52.A 52.A 25.A 25.A 25.A 52. A 52.A 5?. A 0.029 0.017 0.017 0.007 0.010 0.020 0.015 0.016 0.015 0.020 0.015 0.050 0.039 0.052 O.OA'j 0.0A7 0.058 0.088 0.09A 0.093 0.117 0.099 U.G88 0.09A 0.085 0.1 3 0.090 0.131 0.127 0.13? y, - io 0.137 0.109 0.151 0.1 J3 0. 109 0.136 0.161 0.153 y, - 5 P2 - 2 0.187 0.186 0.171 0.209 0.201 0.185 Values for y^ and y^ expressed In Incites. ------- TABLE 5. VALUES FOR DRAINABLE POROSITY CONSTANT, r~C Typ« of Sand Slope (2) Drain Lftigth fft) DFC Fine 2 25.4 0.16 Fins 10 25.4 0.14; Fine 2 52.4 0,151 Fine 10 52.4 0.14i Coarse 2 25.4 0.21! Cjar^e 10 25.4 0. 191 Coarse 2 52.4 0.22< Coarse 10 52.4 0.20( 54 ------- TABLE 6, DRAINAGE TIMES* Tiae co Drain from y, to v (hrs) i * i Type of Sand Slope (2) Drain Length c-:t) v, • 18** y2 - 12 y, - 12 y2 - 6 yl " ' y2 - • Fine 2 25.4 11.0 - - Fine 5 25.4 10.5 16.5 - Fin* 10 25.4 8.0 15.8 - Fin* 2 52.4 24.4 >50 - Fin® 5 52.4 21.6 >37 - Fine 10 52.4 16.2 - - Coarse 2 25.4 7.0 12.1 20.6 Coarse 5 25.'. 4.3 9.2 10.9 Coarse 10 25.4 - 5.0 6.2 Coarse 2 52.4 19.0 36.5 60-5 Coarse 5 52.4 8.7 21.5 27.0 Coarse 10 52.4 12.0 i 3.9 * Drainage tinea la this cable were cakes from 6-hr rainfall experiments, exceet for che fine sand, 2-percenc slope, 25.4-ft drain langth case which was taken from a 24-hr rainfall experiment. Sashes indicate that data were cot available^ ** Values for y ^ and expressed In inches. 55 ------- SECTION 6 VERIFICATION OF THE HELP MODEL Comparisons between the HELP model predictions aaJ Che actual measure- ments are made ia this section to assess the accuracy of the HELP lateral drainage equation. All parameters required to compute the drainage rate by the kEL? equation except the hydraulic conductivity were measured for each drainage test. Hydraulic conductivity measured in a penseaaeter in the soils laboratory differed significantly from hydraulic conductivity calculated from data on drainage rates and depth of saturation measured in the large-scale physical models. This is thought to be due to differences in placement, com- paction, and preparation of the sand drainage media, and due to entrapment of air in the sand. As described in the documentation report for the HELP model (1), the lateral drainage equation vas developed to approximate numerical solutions of the Boussinesq equation for one-dimensional, unsteady, unconfined flow through porous »".«ia (.5). Therefore, the actual hydraulic conductivity for the drainag? rests was estimated by adjusting its value while solving the Boussinesq equation until the results matched the measured drainage rates and saturated depths. The hydraulic conductivity estimates- are summarized in Appendix A. Determining the hydraulic conductivity in this manner provided the best estimate obtainable for each test since the Boussinesq solution is the commonly accepted representation of the actual drainage process. Com- parisons were made for both steady-state drainage during rainfall and unsteady drainage following cessation of rainfall. Comparisons between the Boussinesq solution predictions and the actual measurements are made in the Appendix B to determine how well the most com- monly used theoretical representation of the lateral drainage process per- < forms. The numerical solution of the Boussinesq equation used in this study was developed by Skaggs (5). In addition, the Boussinesq solution predictions are compared with the HELP mcdel predictions in Appendix C. These comparisons are very important because they defln«t the limits of modeling and demonstrate how well the HELP equation represents tl.e Boussinesq solution. COMPARISONS BY HYDRAULIC CONDUCTIVITY First, the results of the steady-state cases are described. The rainfall rata was adjusted until the average head on the lintr (which is also the aver- age depth of saturation, y) reached a_steady-statc height of approximately 12 in. Using this measured value of y and the corresponding measured value of the steady-state drainage rate, the hydraulic conductivity was computed explicitly usiag the HELP drainage equation (Equation 1) with the slope and drainage length employed ia the drainage test. To compute the hydraulic con- ductivity using the Boussinesq solution, a series of numerical Boussinesq runs 56 ------- were made changing the hydraulic conductivity until the computed steady-state y having the measured steady-state drainage rate matched the measured value. A comparison of these values is shown in Table 7, Overall, the computed hydraulic conductivity using the HELP equation exceeded that using the Boussinesq solution by an average of 44 percent. Consequently, the HELP sodel underestimates the drainage rate by 30 percent as predicted by the Boussinesq solution under steady-state conditions. For fine sand, computed values of hydraulic conductivity by the HELP equation exceeded the measured valuss using peraeaaeters by an average of 31 percent; for coarae sand, the average com- puted value was 10 percent of the measured value. TABLE ?. COMPARISON OF COMPUTED HYDRAULIC CONDUCmiTirS FOR STEADY-STATE DRAINAGE Hydraulic Conductivity Rainfall Type of Slope Length Rate a* 3* Tj Sand (X) (ft) (in./hr) (cm/sec) (cm/sec) (cm/sec. Fine 2 25.4 0.034 0.008 0,006 0.004 Fine 10 25.4 0.054 0.006 0.004 0.004 fine 10 52.4 0.025 0.006 0.004 0.005 Coarsa 2 25.4 0.071 0.021 0.015 0.220 Coarse 2 52.4 0.033 0.030 0.021 0.220 Coarse 10 25.4 0.169 0.025 0.015 0.220 Coarse 10 52.4 0.105 0.027 0.021 0.220 * Computed hydraulic conductivity using the HELP equation. ** Computed hydraulic conductivity using the numerical Boussinesq solution, t Hydraulic conductivity measured in laboratory permeaoecers. For unsteady drainage following cessation ef rainfall, hydraulic con- ductivity was computed explicitly_using the HELP drainage equation for instantaneous measured values of y and drainage rate. To compute hydraulic conductivity using the Boussinesq solution, the following steps were followed: (a) Unsteady drainage from the physical model was simulated using the numer- ical Bcussinesq solution with an estimated hydraulic conductivity; (b) at a given value of y, a correction factor was computed by dividing the measured drainage rate by the sinulated drainage rate; and ------- conductivity. Hydraulic conductivities were computed by these two methods at various values of y for each experimental case. Overall, the computed hydraulic conductivity using the HELP tquation exceeded that using the Boussinesq solution by an average of 13 percent. Consequently, the HELP equation underestimates the drainage rate by 11 percent as predicted by the Boussinesq solution under unsteady conditions. These results were determined using a paired-sample t test to determine whether the differences in hydraulic conductivity values between the two methods were sta- tistically different from zero. Table 8 summarizes these results. The test using all data from all experimental cases concluded that the two methods did produce hydraulic conductivity values that were statistically different. Similarly, the tvo methods produced statistically different hydraulic conduc- tivities using the data for only coarse sand or only fine sand; for only short model or only long model; and for oi.ly 2-percent slope or 10-percent slope. However, when the test was conducted for individual experimental cases, the test concluded that in five out of eight cases, the two methods produced hydraulic conductivity values chat were not statistically different. The five cases were: coarse sand in both physical models at 2-percent slope, coarse sand in the long model at 10-percent 3lcpe, and fine sand in both models at 10-percent slope. For both steady and unsteady drainage, the hydraulic conductivity com- puted by the HELP model was shown to exceed that computed by the more rigorous Bous3inesq solution. Consequently, the HELP model might be expected to under- estimate the drainage rate from a drainage system where porous media proper- ties were accurately known. However, in field applications, the magnitude of this underestimation would probably be much less than the uncertainties in the in-place porous media properties. This difficulty was highlighted in this study b> the large differences between laboratory measurements using permease- ters and apparent in-place hydraulic conductivities. Also, the difference between the two solution methods was much less for the unsteady case—the case which generally occurs in field systems. The commuted hydraulic conductivity of the sand during steady-state drainage by the Boussinesq solution was about 61 percent of the value during unsteady drainage. Consequently, if the hydraulic conductivity for unsteady drainage was used in the HELP model for all types of drainage conditions, the lateral drainage rate would be underestimated by II percent during unsteady drainage and overestimated by 14 percent during steady-state drainage. EFFECTS OF ENTRAPPED AIR In analyzing the computed hydraulic conductivity values, it wag apparent that hydraulic conductivity decreased with increasing v. Table 4 shows that the drainable porosity aLso decreased with increasing y. Lower drainable porosity could have resulted by two mechanisms: greater air entrapment or greater compaction. Compaction was 97 percent of maximum throughout the i,amd layer; therefore, air entrapment must have caused the decrease in drainable porosity. Lower drainable porosity results in lower cross-sectional area contributing to drainage; consequently, the hydraulic conductivity decreases. 58 ------- TAHI.K 8 COMI'AKI SON O? COMPUTED HYDRAULIC CONDUCTIVITIES FOR IJNSTKADY DHAINAC.I Standard Statist leal \|Jrt C Mean Value DevlatIon i f;n 1 f 1 cance no • or of (K„ - K ) of (K - K„)t ttt t 1 >( (Ku " ..) Data B ll' B II a/2 B II Data Croup Points** (cm/sec) (cm/sec) Stat If tic (a ¦= 0.05) at a - 0.05 All data 93 -0.00309 0.00693 -4.300 -1.960 Not equa 1 Li> zero C 43 -0.00589 0.00943 -4.098 -1.963 Not cqu* 1 to zero F 50 -0.00608 0.00101 -4. 740 -1 .960 Nor e<]u;i 1 In zero 2 51 -0.00083 0.00135 -4.375 -1.960 Not equal t (i zero 10 30 -0.00584 0.01055 -3.033 -2.045 Not equa 1 to zero S 49 -0.00314 0.00815 -2.697 -1 .960 Not equa 1 to zero L 44 -0.00303 0.00534 -3.765 -1.960 Not equal to zero C2S 10 -0.00067 0.00107 -1 .981 -2.262 M-« > equal zero C10S 8 -0.01609 0.01491 -J.053 -2.365 Not equal to zero C21. II -0.00104 0.00233 -1 .479 -2 .228 May eqml zero C10L 6 -0.00749 0.00746 -2.459 -2.571 May equal ze i o F2S 17 -0.00054 0.0004 7 -4.782 -2. 120 Not equa 1 to zero F10S 8 -0.00031 0.J0073 -1.222 -2.365 May equal ze ro F2I. 13 -0.00114 0.00127 -3.235 -2. 179 Not equa I to zero F10I. 8 0.000!? 0.00061 0.552 2.365 May equa 1 ze ro * C - coarse aand; S - short model; 2 - 2Z slr.pe. F ¦ Fine band; I. - long model; 10 = I0Z slope. ^ Ea jh data point represents one (K - K^) value at particular y. t Kp - Computed hydraulic conductivity using the numerical Houeulnesq solution, tt K = Computed hydraulic conductivity using the IIELI' equation. H t - where x - mean value of (KR - . s//n Mu - hypothetical population mean ~ 0. 8 - standard deviation of (K^ - K((). n «=• number of data points. ------- To determine whether variability in the estimates of the hydraulic con- ductivity was a function of other variables in addition to average saturated depth. a series of unequal three-way analysis of variance (ANOVA) tests was run cm the rseth sets cf estimates. This procedure permits analysis of experi- ments that have three variables called treatments, which produce variance in the neasured variable. AKOVA also permits determination of interaction between variables. The variance »nd the number of duplicates cf treatments die! net have to be equal in the prot-tdure. The analyses were performed using Che Statpro software by Penton Software Inc. (6). The test variables include type of sand, average saturated depth, s»lope, drainage length, rainfall duration and rainfall intensity. No effects cf rainfall duration and intensity could be discerned by inspection: therefore, the initial ASOVAs were run using depth, slope and length as the variables for data sets containing hydraulic conductivity estimates for one type of sand. These ANDVAs indicated that the hydraulic conductivity estimates by HELP varied only as a function of average saturated depth for coarse sand and of average saturated depth and slops for fin* sand. Running ANCVAs on the estimates generated by the Boussinesq solution, the estimates varied significantly with depth and slcpe for both sanir. No physical reasons are apparent for the variability of the hydraulic conductivity as a function of slope. Therefore, the variability arises from inaccuracies in the manners in which the effects of slope are modeled by the Boussinesq equation and the HELP equation. This difference is presented ia greater detail later in this section and is Appendices 3 and C. A regression analysis wis conducted to investigate the relationship between hydraulic conductivity and y. Evdraulic conductivity, K, was fitted to a power function of y such that K - a yfc 0 5) where values of the regression coefficients a and b are listed in Table 9 for 10 of the experimental cases. This equation applies only to y values ranging froa 2,5 to JO in. in these drainage tests. In Equation 15, the units for K are centimeters per second and the units for y are inches. The relationship between drair.able porcsity and height above the liner is presented in Section 5. COMPARISON OF DRAINAGE RATES To further test the accuracy of the HELP lateral drainage equation, the relationship between drainage rate and y as predicted by the HELP drainage equation was compared to measured results for unsteady drainage following ces- sation of rainfall. The ccaparisons are shown in Figures .25 and 26. Measured results are represented by the shaded area which defines the range of all mea- sured data for the given sand, slope and drainage length. Three computed curves are also plotted. One curve represents the HELP drainage equation using the hydraulic conductivity measured in the laboratory permeaseters. A second curve represents the HELP drainage equation using the mean of hydraulic conductivity estimates obtained froo the Boussinesq solution for all unsteady drainage Jests performed with the given sand, slope and length. The third 60 ------- TABLE 9. REGRESSION COEFFICIENTS FOR HYDRAULIC CONDUCTIVITY AS A POWER FUNCTION OF y* Type of Sar.d Slope Drain Length (ft) a b T ine 25,4 0.0951 -0,6650 Fine 10 25.4 0,0243 -0.6166 Fine 52.4 0.067? -0.7252 Fine 10 52.4 0.0048 0.0964 Coarse 2 25.4 0.1297 -0.6934 Coarse 5 25.4 0.0675 -0.4359 Coarse 10 25.4 0.0477 -0.2392 Coarse 2 52,4 0,1057 -0.5383 Coarse 5 52,4 0.0313 -0.2444 Coarse 10 52,4 0.0395 -0.1896 * The regression equation is K « a y where K - hydraulic conductivity ir. centimeters per second, y ¦ average depth of saturation or head on the liner in inches, and a,b - regression coefficients, curve also represents the HELP equation but uses Equation 15 for hydraulic conductivity. The curves based on laboratory measurements of hydraulic con- ductivity resulted it poor fits except for the fev cases where the measured hydraulic conductivity was ciose to the computed values, "hu again high- lights the difficulty in .e-j'inating in-place hydraulic conductivity from laboratory measurements. The curves based on a mean value of hydraulic con- ductivity fit reasonably well within a _-latively narrow band representing the y region for which the Bean hydraulic conductivity was computed. Outside this band, the predicted curves tended to deviate, especially for the short aodel. The curve b~sed on Equation 15 generally fit well for y values lets than 15 to 20 in. for the fine sand and leas than 12 to 15 in. for the coarse sand. These also represent the y ranges for which Equation 15 and Table 9 were derived, Mnra of the predicted curves fit wall at y values close to zero due to r.he measured capillary water drainage which the HELP equation does not model. 61 ------- Pradicivd fUwItii « (it k O) • MOftt MOOtl «% IIOM AVERaU IICAO (inchtfs) Pr«dicld ffssullai • N O fttOT I Ui ~— or lONO MOOCi < rr D AVEHAGl Hi AO (melius) Pr«dKi») K«iu\|ii4 lab K O fatf. M cn 10% tlO'I i; rr j AVIKAGI IK AO (locties) A-i Predicted Riiullu • lab K li Sweat H I 15- £ •< a: lONO MOOfl to% llOM oc n 1/ ~l6 Si AVIRALC lit AO (inches) Figure 25. heasured drainage rate following rainfall vs. average head niiuve the liner compared to HELP drainage equation predictions for t'lne sand In both physical models at slopes of 2 and 10 percent. ------- AVERAGE ItfAU (inLhwb) AVERAtl IAD (inchas) AVLRAW ItLACI (inches) AVEKACl Ml All (inches! Figure 26. Measured drainage rate following rainfall vs. average had above the liner compared to HELP equation predictions for coarse sand In both physical models at slopes of 2 and 10 percent. ------- SIMULATIONS ST REL? MODEL Figures 27-30 shew hew veil th ------- ?« !-•m itAiM ai ii 14 ttt tm SHQ8I wnotl A! 21 UP\| I -c a M ( D M Mill 1 0. 57 IN /IN MUHI MUX. 1 131 a« .a fc? £ 3 i lo IT la k <• IMC Qio>j"s! a* VJ> ?« !-*> KAN » U H It /m W0f «am »r /i m»c mimiiu a AC IIJU X I (. 0 M Ml* il 1:1 m /M SMCR* MML 1 10. si arc "Uo°o?L"r-~— "••to o II- y I p.ure 27, Htreurtd drainage rate and average sa' urat pd depth vs. tin' compared to IIKI.." j/rt ill r». ion for fine sand In the aliort model at slopes of 2 and 10 percent. ------- 1.2 ** A\|M 1 0 IN, /MM ICNC A! (Ill SiUPf - mcoicicfi t* ACIU41 8 I' . 4.m iuim ai b.si m Am itxr> MOQii i n Siort - PMLOIC 1(0 o KIUM. .12 § ------- i i «: RMM ai (i ^ in a« fcOfl «£i! 1 Idl S»tft - PHU#uiia O A« IUAL i a> 3 V la h It I U-m RAIN A? 0 • l/Ht Mim ai ?i stcrt - mmuio a ACruAl j. « ta la I iHt Oiourt) I Jfi£ Owiurfc) n Figure 8 o hi rain hi a. si in /iti SiiCRT nnBft it MOPt PWOIUIQ KIUAI 'V^ 1r-» £ £ II ------- 2 m RAlH 1 0 57 IN />H LUNC M0UT1 Al IQ1 SI - PNtOICTIO u ACTUAL ft RAIN At 0 U IN /M I QNt MUCH.I Al 21 SLGPl i Ui or z •< S 0- 00 6 I-KM RAIN Al 0 » IN /Mt 10NG WOfL Af ?X SLOPt — pwntcrio o AC1U4L ^-4-1 Jo 2a TlHo (htac) 5 £ & t ------- .25 6.2-WR SAIN AT 0.54 IN. ."Hfl — PSEDICTED a ACTUAL ! m UJ =c S B .05- 6, 2-HR SAIN AT 0. 54 IH. Ati Ui I c a. a 10-1 TIME (hours) Figure 31. Measured drainage race and average saturated depth vs. time conpared to HELP prediction for fine sand in the long model at i0-percent slope using oean values for hydraulic conductivity and drainable porosity. 69 ------- & 1-HR BAIN AT 0.56 IK/m m Q_ en .3- z 8 TIME (hours) 6, l-W RAIN A? 0.56 IN./MR UJ at Figure 32. Measured drainage race and average saturated depth va. tia« compared to HELP prediction for coarse aand in the short model ac 10-percent slope using mean values for hydraulic conductivity and drainable porosity. 70 ------- created as duplicates in this analysis. First, th« effect of drainage length was isolated by using Equation 1 to compute the ratio of drairage rata for the long model (Q^) to drainage rate for the short model (Q ) given the sane drainage media and constant values for a and y, This fitio was tultipli-'d by the measured Qg to predict Q, , The difference betveen measured and predicted was assumed to be due to a combination of experimental error and simplify- ing assumptions used in the derivation of the HELP drainage e^uatios. Table 10 summarizes these comparisons, and Figure 33 plots the predicted versus measured Q^. A regression analysis was conducted using Che data in Table 10 and Figure 33 to find the slope of the best-fit line vhich also passed through the origin. The regression analyses performed to evaluate the HELP equation are summarized in^Table 11. The resulting slope was 0.82 with a coefficient of 1#terminal!ion (r ) of 0.99. The 95-percent confidence interval for the slope did not include the value of 1.00. The slope should equal 1.00 and Che intercept should equal 0.0 if the predicted and actual results are identical and the predictive method is accurate. Therefore, Qf as measured in the unsteady drainage tests and as predicted by the HELP equation based on a measured Q are statistically different. The HELP lateral drainage equation overestimates the decrease in drainage rate resulting from an increase in length given the same sand, slope, and y. The effect of slope was similarly isolated by using Equation 1 to compute the ratio of drainage rate for the 10-percent slope (Qirj?) to the drainage race for the 2-percent slope (Oj) given the sane sand and constant values for L and y. This ratio was multiplied by the measured Q,. to predict Q,. Table 12 summarises these comparisons, and Figure 34 plots the predicted Q,a» versus measured Q-0_. A regression analysis was conducted using the data In" Table 12 and Figure 34 to find the slope of Che best-fit line wtjich also passed through the origin. The resulting slope was 1.30 with r* - 0.98. The 95-pircer.t confidence interval for Che slope of this li*e did not Include the value of 1.0C. Therefore, Q^q« as measured in the laboratory and as predicted by the HELP equaLion based on a measured are statistically different. The HELP lateral drainage equation overestimates the increase ia drainage rate resulting from an increase in slope. Similarlyi Equation 1 wus used to compute the ratio of drainage rate for y • 12 in. CQI?„) to drainage rati fer y - 6 in. (Q^») given the sane sand ar.d conscar.t values for L and i. In the calculation of this ratio it was assumed Chat the hydraulic conductivity at y - 12 in. was equal Co 0.732 times the hydraulic conductivity at y • 6 in. based on Equation 15. This ratio was multiplied by the measured Q,,, to predict Q, . Table 13 summarizes these comparisons, and Figure 35 plots the predic?ed Q.," versus neasured Q , T\|e slope of the best-fit line passing through trie origin was 1.02 with r - 0.99. The 95-percent confidence interval for the slope of this line included the value 1.00. Therefore, Q^" as measured in the laboratory and as predicted by the HELP equation based on a measured Qg„ are not statistically different. The HELP lateral drainage equation accounts for the effect of changes in y very well. In the HELP lateral drainage equation (Equation L) the slope (a) and drainage length (L) occur as a product several times. This product physically represents the Head above the drain contributed by Che slope of the liner. 71 ------- TABLE 10. EFFECT OF LENGTH IN THE HEL? DRAINAGE EQUATION Q./Q Measured Measured Predicted** Type of y Slope frots H§L? Q Qt Q. Sand (la.) (I) Ecuador, (gpg) (gpfl) (gpfi) Fine 6 2 0 737 0 030 0.025 0.022 8 2 0 689 0 044 0.030 0.030 5 0 868 0 043 0.040 0.037 10 1 005 0 058 0.061 0.058 12 2 0 626 0 056 0.039 0.035 5 0 802 0 058 0.051 0.047 10 0 956 0 072 0.070 0.069 14 2 0 604 0 067 0.045 0.040 5 0 775 0 068 0.057 0.053 10 0 933 0 082 0.081 0.077 Coarse 6 2 0 737 0 056 0.052 0,041 5 0 909 0 098 0.094 Q .089 10 1 033 0 164 0.166 0.169 8 2 0 689 0 070 0.067 0.048 5 0 868 . 0 1.-3 0.116 0.107 10 1 005 0 183 0.204 0.184 12 2 0 626 0 119 0.106 0.074 j 0 802 0 177 0,169 0.142 10 0 936 0 244 0.287 0.233 14 2 0 €04 0 151 0.139 0.091 5 0. 775 0 218 0.210 0.169 10 0. 933 0 264 0.334 0.246 * ¦ Drainage rate froa long model. Q - Drainage rata froa short model, s + Prtdictpd Q, ¦ (Q, /Q frum HELP Equation) x (Measured Q }. L L S S 72 ------- .25- 15" . 1 ^ FINE SAND, y a FINE SAND, y FINE SAND, y COARSE SAND. COARSE SANU. COARSE COARSE SAND, 1 T .1 .15 .2 ACTUAL DRAINAGE RATE Figure 33. Predicted vs. actual drainage rate UBlng measured drainage from short model to predict drainage from long model. ------- TABLE 11. REGRESSION ANALYSES SUMMARY FOR EVALUATION OF THE SEU1 LATERAL DRAINAGE EQUATION 952 Conftderce 1laics Data Set 0 /C V \v Vqa Intercept N** 2 r Effect of length, constant a 0.819 0.749 to 0.890 -0.0070 to 0.0125 22 0.987 Effect of s^ooe 1.302 1.086 to 1.519 -0.0269 to 0.0511 15 0.979 Effect of y 1.023 0.866 to 1.193 -0.0207 to 0.0279 10 0.989 Effect nf aL, cons taur L 1.190 1.115 to 1.276 -0.0099 to 0.0130 15 0.996 * Qu - Drainage race predicted by HEL? equation. Q" - Drainage race measured in drainage test. ** fj* « number of values In data set. Therefore, the effect of aL was similarly Isolated by using Equation 1 to com- pute the rzzlo of drainage rate cor an aL value of about 30.5 in. (Q,n ,«,) t.o drainage rate for an aL value of about 15.3 in. (Qic ¦,.¦) » given the same 3and and constant values of L and 7. This ratio was multiplied by tha asasursf Q.j t0 predict j,.. Table 14 summarizes these comparisons and Figure 36 plots the predicted versus aeasured Q.^ A regression analysi-j was conducted using the data in Table 14 and Figure 36 to find the slupc ?£ Che best-fit line which also passed through the origin. The resuliing slopi 1.20 with r • 0.996. The 95-percent confidence interval for the slo>r did not include the value of 1.00. Therefore, Q-- as measured in the labora- tory and as predicted by th. HELP equaticn bastd on a measured Q . are sta- tist ically different. The HEL? lateral drain;3e equation overestiaatis :ha increase in drainage rate resulting from an increase in slope. Earlier, the effect of length was isolated frota the effects of slope, type of slope, type of sand, and y, but it was not isolated from the effect of aL. To isolate cbe effects of length from the effect of aL, the rauio of drainage rate for the long model (0r) to the drainage rar.e for the short model (Q ) given the same sand and constant values of y and aL was computed using Equation I. This ratio wis multiplied by the aeasured Q to predict 0., Table 15 summarizes these comparisons and Figure 37 plot!' the predicted Q versus measured Q . A regression analysis was conducted using the data in Table 15 and Figufe 37 to ------- TA3LE 12. EFTECT Of SLOPE IN THE HELP DRAINAGE EQUATION Type of Sand L (In.) V (in. ^ 10Z ^2Z from EELP ) Equation Measured q2Z (gpta) Measured Q10X (g?a>) Predicted** °102 (pa) F ine 305.4 6 3.187 0.030 0.050 0.096 8 2.804 0.044 0 .058 0.123 12 2.300 0.056 0.072 0.129 14 2.12b 0.067 0.082 0.143 629.4 8 4.090 0.030 0.061 0.123 12 3.512 0.039 0.070 0.137 14 3.287 0.045 0.081 0.148 Coarse 305.4 6 3.187 0.056 0.164 0,178 8 2.804 0.070 0.183 0.196 12 2.300 0.119 0.244 0.274 14 2.128 0.151 0.264 0.321 629.4 6 4 a 464 0.052 0.166 0.232 8 4.090 0.067 . 0.204 0.274 12 3.512 0.106 0.287 0.372 14 3.287 0.129 0.334 0.457 0 - ? Y Q m yioz Drainage Drainage race race from 2-percent from 10-percent slope. slope. ** Predict ad Q10% * CQlOj'/Qi« fr0D HELP Equation) x (Measured "zx5' 75 ------- E C_ cn c Cd LD ¦C en a CD <_> a LlJ cr Q_ 305 305 305.4 305.4 629. 4 629.4 629. 4 629.4' .08 ACTUAL . 12 DRAINAGE Figure 34 Predicted vs. actual drainage rate using measured dralnape at 2-percent slope to predict drainage at 10-percent slope. ------- TABLE 13. EFFECT OF AVFRAGE DEPTH OF SATURATION IN THE HELP DRAINAGE EQUATION* Q, Vi/Qfi'i Measured Measured Predicted** Type of L Slooe from HELP Q6" Q. Ql2" Sand (in.) r.) Equation (gp®) (gpm) (Spn) Fine 305.4 i. 2.357 0.030 0 056 0.071 5 1.918 0.032 0 058 0.062 10 1.693 0.050 0 c;2 0.085 529.4 5 1.99S 0.025 0 039 0.050 Coarse 303 .4 2 k .357 0.056 0 119 0.132 r 1.918 0.098 . 0 177 0.188 10 1.698 0.164 0 244 0.279 679.4 2 1.998 0.052 0 106 0.104 5 1.695 0.094 0 168 0.160 to 1.574 0. 166 0 237 0.261 Q^., - Drainage race f>om v » -> in. Qt1«i " Drainage rate from v » i 2 in, ** Predicted Q,»„ ¦ (C, -»«./Q .. from HELP ciuacion) x '.Measured I c I i. 0 h 77 ------- E CL cn Ck L = 3()rj. 4' , « -= 77. ~ L - 305.4' . « - M o 1 - 305.4' . a = m A L - 629.4' .. « - 21 ¦ L = 629. 4' . a = b I • L = 629.4' . «= m ¦c CK CD ¦C ¦c an CD n (_j a U) oc Q_ T 1 .2 ACTUAL DRAINAGE-: RATE (gpm) Figure 35. Predicted vs. actual dralnag. rate using measured drainage «t y - 6 in. to predict drainage at y * 12 In. ------- TABLE 14. EFFECT OF aL IN THE HELP DRAINAGE EQUATION* Type of L Sand (ia.) y (ia.) Q30.5"/Q15.3" from HELP Equation Measured Q15.3" (g?ns) Measured Predicted** Q30.5" ^30,5" (Spm) (gpn) Fine 305,4 6 1.815 0.032 0 050 0.058 8 1.736 0.043 0 058 0.075 12 1.609 Q. 058 0 072 0.093 14 1.557 0.36<3 0 082 0.106 629.4 3 1.736 0.034 0 039 0.059 12 1.609 0.043 0 050 0.069 14 1.557 0.049 0 056 0.076 Coarse 305.4 6 1.815 0.098 0 164 0.178 8 1.736 0.123 0 183 0.213 12 1.609 0.177 0 244 0.284 14 1.557 0.218 0 264 0 339 629.4 6 1.815 0.060 0 091 f .109 8 1.736 0.076 0 113 0.132 12 1.609 0.117 0 163 O.188 14 1.557 0.152 0 205 0.237 * Q,0 • Drainage Qj2'3» " Drainage rate rate for aL • 30.5 in. for aL - 15.3 ia. ** Predicted ¦ CQ30 5"^L5 3" ^ro® Equation) x (Measured Qjj 3«). 79 ------- CD o E d_ CD Ul Q1 11J LD < >—* < cr. tn en UJ »- - CD iii l YL n_ 305.4 305.4 305.4 305.4 629.4 G In. 15.1 In. ------- TABLE 15. EFFECT OF LENGTH IN THE 3EL? DRAINAGE EQUATION GIVEN CONSTAiVT sL* Q, >'Q. Measured Predicted** Type of Sand aL (in.) V (in.) :• ~~ 3 \ .i_an 9s (po) ql (g?m) Q, L, (gpm) Fine 15.1 6 0.485 0 032 0.U29 0.016 Fine 15.3 3 0.485 0 043 Q.C34 0.021 Fine 15.3 12 C.485 0 058 0.043 0.028 Fine 15.3 14 0.if.5 c 068 0.049 0.023 Fine 30. S 8 0.485 0 053 3.039 0.028 Fine 30.5 12 0.-35 0 0"2 Q.05C 0.035 F ine 3C.5 14 0.435 0 062 0.056 0.040 Coarse 15.3 6 0.485 c 098 0.060 0.048 Coarse 15.3 8 0.485 0 123 0.076 0 060 Coarse 15.3 12 0.485 0 177 0.117 C .086 Coarse 15.3 14 0.485 0 218 0.152 0.106 Coarse 30.5 £ 0,i85 0 164 0.091 0.030 Coarse 30.5 8 0.485 c 183 0.113 0.039 Coarse 30.5 12 0.485 0 244 0.163 0.113 Coarse 30.5 14 0.485 0 264 0.205 0.128 Qj_ " Drainage rate from Long model. Q » Drainage race from shore model, s ** Predicted Q, ¦ (Q. /Q from HELP equation) ;< (Measured Q ). L L 3 S 81 ------- «L «L - 15.3 "L = 15.3 «L = 3D. 5 «L = 30.5 «L = 30.5 «L = 30.5 E Q_ CD .15" Lul LU CD ¦C ~z. I—I a LU 5 .05- a LxJ Ctl Q_ .05 ACTUAL DRAINAGE RATE (gpm) Figure 37. Predicted vs. actual drainage rate using measured drainage from short to predict drainage from long model having the same value of al.. ------- SECTION 7 SUMMARY AND CONCLUSIONS Drainage testa were performed on cvo large-scale physical models of land- fill liner/drain systems Co examine Che effeces thac che length, slope and hydraulic conductivity of the drain layer and the depth of saturation above the liaer have on Che subsurface lateral drainage rate. The models have dif- ferent lengths. 25.4 ft and 52,4 ft, and adjustable slope rangirg from 2 to 10 percent The models were filled wich a 3-ft sand drain layer overlying a l-ft clay liner, A 2-in. layer of gravel was placed under Che liner to collect seepage from the clay liner, Drainage from tvo different sands was examined it, both models. • Several drainage tests wer? run on each configuration of the models by applying water as r?infall Co che surfacn of the sand layer and Chen measuring the water cable along the lengch of the models and Che lateral drainage volume as a function of time. Lateral drainage and water table profiles were mea- sured during periods of increasing, decreasing, and stetdy-stace drainage rates. In total, more Chan 60 cescs were performed. The hydraulic conductivity of the tand was measured in the laboratory permeaaecers, but its value was apparently quite different and varied in the drainage testa. The hydraulic conducel\ity of the sand ac various depths cf saturation was estimated for each test vsing the Bousslnesq solution of Darcy's lav for unsteady, unconfined flew through porous media and the HELP lateral drainage equation, These hydraulic conductivity values were compared to deter.nine the agreement between the HELP model and Che Bousslnesq solution. For steady-state drainage the HELP modtl estimates of the hydraulic ccnuuc* Civity were 44 perenne greater than Che Boussinesq solution estimates, Th'.j result means that the HELP model underestimated che steady-state latera1 drainage rata predicted by the Bousslnesq solution by 30 percent. The HELP estimates were 31 percent greater than the laboratory measurements for the fine sand and 90 percent less than the laboratory measurements for the coarse sand. For unsteady drainage the HEL? model estimates were only 13 percent greater Chan the Boussir.ssq solution estimates, which would underpredict the lateral drainage rate by 11 percent. The underpreJiction of the cumulative latetai drainage volume would be expected to be very small since che removal rate of vater from the drain layer by all other means is much smaller than the lateral drainage rata. Consequently, the effect of differences in the pre- dicted and actual drainage Clmea i« small. The differences between the laboratory measurement of the hydraulic con- ductivity and either of the tvo estimates were much larger than the differ- ences between Che estlaaCes. In addition, the hydraulic conductivity varied 83 ------- — _ ^ wae uejjcn or saturation, apparently due to entrapment of air in the sand. Similarly, the dralnable porosity varied as a function of depth of saturation due to the air. These phenomena cake it very difficult to model the lateral drainage process and produce good agreement becveen the predicted and actual results for drainage rate and depth of saturation as a function of t i line . An analysis was performed to determine how weJ1 the lateral drainage equation in the HELP modal accounts for the effects of drainage length, slope of the liner, average depth of saturation and head above the drain contributed by the liner (aL) in the estimation of the drainage rftte. The drainage equa- tion overestimates the decrease in drainage rate by an increare in length given the same sand, slope, depth of saturation and >ead £.lov# the drain, Similarly, the equation overestimated the increase in drainage rate resulting from an increase in slope and head above the drain. The effects of dept'. of saturation predicted by the equation agreed very well with the actual results. The following conclusions and recommendations are made. Lateral drainage in landfill liner/drain systems is quite variable, probably due to air entrap- ment. The hydraulic conductivity measurement made in the laboratory is quite different than the in-place value. Consequently, the estimation of the lat- eral drainage race is prone to considerable error despite having a good equa- tion or solution method for the estimation. Nevertheless, the prediction of the cumulative volume of lateral drainage is likely to be quite good since the depth of saturation will be overpredlcted if the drainage rare is underpre- dicted and vice versa, thereby adjusting the drainage rate. However, the pre- dicted deprh of saturation will be quite different from the measured value. The lateral drainage equation in the HELP model performs very well for tests on models at 2-percent slope but overpredlcted drainage and depths at 10-percent slope. The model overestimates the effect of drainage length in reducing the drainage rate and also overestimates the effects of slope and head above the drain in increasing the drainags rate. Therefore, additional refinement in the equation should be performed to make the equation valid over a wider range of slopes and drainage lengths. Nevertheless, the HELP equation does provide a good estimation of the lateral drainage volume. Evaluation of the effects of drainage length, slope of the liner, depth of saturation ar.d head above the drain on the drainage rate predicted by the Boussinesq solution should be performed to determine whether the effects observed with the FEFLP drainage equation are unique. Similarly, an additional data set of drainage results should be collected to determine vhether the effects are unique to this data set. Additional data should be collected for longer drainage lengths and greater slopes and from actual landfill/liner systems. 84 ------- REFERENCES 1. Schroeder, P. R.» Gibson, A, C.. and Ssclen, M. D, The Hydrologic Evaluation of Landfill Performance (HELP) Model, Voluae II. Documentation for Version I. E?A/530-SV-€4-0l0, U.S. Environmental Protection Agency, Offict of Solid Waste and Emergency Response, Washington, D.C, 1984, 256 pp. 2. Schroeder, P. R.» Morgan, J. M,, Wtlski, T. M., and Gibson, A. C, The Hydrologic Evaluation of Laadfill Performance (HELP) Model. Voluae I. User's Guide for Version I. EPA/530-SV-S4-C09, U.S. Environaetita1 Protection Agency, Office of Solid Waste and Emergency Response, Washington, D.C., 1984. 120 pp. 3. U.S. Environmental Protection Agency. RCRA Guidance Document for Landfill Design. Federil Register, General Services Administration, Washington, D.C., July 26, 1982. 4. Headquarters, Departnent of the Army, Office, Chief of Engineers. Labora- tory Soils Testing. Engineer Manu/.l H10-2-1906, Washington, D.C., 1970. 5. Skaggs, R. W. Modification to DRAINMOD to Consider Drainage from and Seepage Through a Landfill, Appendix A. N'uoerical Solution to the Boussinesq Equation for Transient Water Movement over an Inclined Restrictive Layer, Draft Report, U.S. Environmental Protection Agency, Cincinnati, OH, 1982. 15 pp. 6. Penton Software, Inc. Stacpro User's Manual. Sew York, 1965, 85 ------- BIBLIOGRAPHY Baver, I. D.» Waiter K, Gardner, and W'ilfcrd R. Gardner. 1972. Soil Physics. John Wiley and Sons, New York. 484 pp. Carlson, E. J. 1971. Drainage from Level and Sloping Land. REC-EXC-71-44, Engineering and Research Center, Bureau of Reclamation, Denver, CO. 43 pp. Dak.i, J. K. K. 1972. Essentials of Engineering Hydraulics. John Wiley and Sons, Inc., New York. 392 pp. Freize, 1. All o, anJ John A. Cherry. 1979. Groundwater. Prentice-Hall, Inc., Er.glevood Cliffs, HJ. 553 pp. Hillel, Daniel. 1971. Soil -,nd Kate.': Physical Principles and Processes. Acadenic Press, New York. 288 pp. Hillel, Daniel. 1982. Introduction to Soil Physics. Acadeaic Press, Mew York. 364 pp. List, E. John. 1965. Steady Flow to Tile drains Above an Impervious Layer - A Theoretical Study. KH-R-9, California Institute of Technology, Pasadena, CA, 60 pp. Luthin, Janes N. 1966, Drainage Engineering. John Wiley and Sons, Inc., New York. 250 pp, 86 ------- APPENDIX A HYDRAULIC CONDUCTIVITY ESTIMATES The analysis performed la Section 6 of this report required estimates of th* hydraulic conductivity of the drainage media. These estimates were generated using the ouaerical Boussiaesq solution and the EEL? oodel lateral drainage equation. Estimates were generated for sost of the drainage tests incindiug steady-state drainage, unsteady drainage following rainfall, and unsteady drainage from presaturated sand. The hydraulic conductivity values were astimated at several depths for the unsteady drainage tests. Values for steady-state drainage, unsteady drainage following rainfall, and unsteady drainage from presaturated sand are given, respectively, in Tables A-l, A-2, and A-3. TABLE A-2. HYDRAULIC CONDUCTIVITY ESTIMATES FOR STEADY-STATE DRAINAGE Model Average Drain Hydraulic Conductivity Type Length Slope Depth Rate (in./hr) of Sand (in.) (X) (in.) (in./hr) Boussinesq HELP Pine 205.4 1.8 12.6 0.0336 0.0061 0.0084 Fine 305.4 10.4 13.2 0.9543 C.0Cj7 0.0055 Fine 629.4 10.3 12.? 0.0249 0.0041 0.0058 Coarse 305.4 1.9 11.4 0.C708 0.015 0.021 Ccarse 305.4 10.0 11.3 0.1688 0.015 0.025 Coarse 629.4 2.3 11.3 0.0330 0.021 0.030 Coarse 629.4 9.9 11.8 0.1050 0.021 0.027 87 ------- TABLE A-2. HYDRAULIC CONDUCTIVITY ESTIMATES FOR ¦JSSTfAD'? DRAINAGE FOLLOWING RAINFALL Model Rainfall Kydraulic Conduc tivity Type Length Slope l>jration Depth (in./hr) f Sand (in.) C) (hr) (in.) Scussines: UCT p Fine 305.4 1.8 1 9 C.0138 0.0147 Fine 305.4 1.8 1 8 0.0151 o.oisa Fine 305.4 1.8 1 7 0.0173 0.0180 Fine 305.4 1.8 2 19 0.0048 0.0063 Fine 305.4 1.8 2 17 0.01 11 0.0107 Fine 305.4 1.8 2 10 0.0200 0.0211 Fine 305.4 1.8 2 10 0.0172 0.0180 Fine 305.4 1.8 6 17 0.0116 0.0117 Fine 305.4 l.S 6 9 0.0182 0.0183 Fine 305.4 1.8 24 19 0,0064 0.0063 Fine 305.4 1.3 24 15 0.0078 0.0080 Fine 305.4 1.8 24 5 0.0212 0.0218 Fine 305.4 :.8 24 4 0.0251 0.0262 Fine 305.4 4.9 6 P 0.0061 0,0069 Fine 305.4 4.9 6 7 0.0106 0.0112 Fine 305.4 10.4 1 6 0.0083 0.0C96 fine 305.4 10.4 1 5 0.0092 J.0109 Fine 305.4 10.4 2 17 O.OC61 0.0059 Fine 305.4 10.4 2 11 0.0067 0.0066 Fine 305.4 10,4 6 16 0.0060 0.0058 Fine 305.4 10.4 6 8 0.0079 0.0082 Fine 305.4 10.4 24 15 0.0039 0.0033 Fine i05.4 10.4 24 12 0.0044 0.0043 Fine 629.4 2.1 1 8 0.0164 0.0165 Fine 629.4 2.1 1 7 0.0179 0.0184 Fine 629.4 2.1 2 20 0.0070 0.0108 Fine 629.4 2.1 2 16 0.0086 0.0104 Fine 629.4 2.1 2 10 0.0132 0.0135 Fine 629.4 7.1 6 20 0.0084 0.0099 Fine 629.4 2.1 6 10 0.0134 0.0136 (Continued) 88 ------- TABLE A-2. (CONTINUED) Type of Sand Model Length fin,) Slope '25 Rainfall Duration Chr) Depth (in.) Hydraulic Conductivity (in,/hr) Boussinasa HEL3 Fine 629.4 2.1 24 19 0,0081 0.0092 Fiat 629.4 2. 1 24 9 0.0128 0.0132 Fin« 629.4 5.0 6 ia 0.0050 0.0075 Fine 629.4 5.0 6 9 0.0073 0.0107 Fine 629.4 10.3 1 13 0.0054 0.0062 Flo* 629.4 10.3 1 11 0.0062 0.0069 Fi»« 629.4 10.3 2 17 0.0061 0.0058 Fine 629.4 1C.3 2 14 0.0057 0.0055 ?ine 629.4 10.3 6 36 0.0068 0.00'il Fin* 629.4 10.3 6 11 0.0064 0.00(3 Fine 629.4 10.3 24 1? 0.0C65 0.0C58 Fine 629.4 10.3 24 13 0.0062 0.0058 Coarse 305.4 1.9 2 15 0.0228 0.0218 Coarse 305.4 1.9 2 10 0.0276 0.0274 Coarse 305.- 1.9 2 4 0.0720 0.078 Cv-arse 305.4 1.9 0 15 0,0225 U.0227 Coarse 305.4 1.9 6 11 0.0279 0.0274 Coarse 305.4 1.9 6 6 0.0413 0.0431 Coarse 305.4 1.9 24 8 0.0317 0.0319 Coarse 30i. 4 1.9 24 8 0.0316 0.0315 Coirse 305.4 1.9 24 5 0.0508 0.052? Coarse 305.4 1.9 24 4 0.0734 0.0745 Coarse 305.4 5.4 2 9 0.0309 0.0306 Coarse 305.4 5.4 2 5 0.0383 0.0409 Coarse 305.4 5.4 6 9 0.0253 0.0251 Coarse 3015.4 5.4 6 4 0.0394 0.0418 Coarse 305.4 10.0 2 6 0.0348 0.0370 Ccars 305.4 10.0 2 2 0.0472 0.0790 Coarse 303,4 10.0 6 7 0.0300 0.0334 Coarse 305.4 10.0 6 2 0.0456 0.0747 (Continued) 89 ------- TABLE A-2. (CONCLUDED) Type of Sand Model Length (In.) Slope (?) Rainfall Duration (hr) Depth (in.) Hydrav.lic Conductivity (in./hr) Bcussinesc HEL? Coarse 305.4 10.0 24 3 0.0301 0. Oil 24 Coarse 305.4 10.0 24 1 0.0431 0.0814 Coarse 6:9.4 2.3 2 16 0.0292 0.0238 Coarse 629.4 2.3 2 11 0.0301 0.0294 Coarse 629.4 2.3 2 5 • 0.0500 0.05C8 Coarse 629.4 2.3 6 11 0.0277 0.0276 Coarse 629.4 2.3 6 3 0.0692 0,0761 Coarse 629.4 2.3 :4 8 0.0354 0.0350 Coarse 629.4 2.3 > 24 £ 0.0553 0.0582 Coarse 629.4 5.2 2 i: 0,0213 C.L 294 Coa-se 629.4 5.2 2 4 0.0281 C.0-.63 Coats* 629.4 5.2 6 10 0.015? G.02S' Coarye 629.4 5-2 e 5 0.0164 0.0349 Coarsa 629.4 9.9 2 9 0.0278 0.0290 Coarse 629.4 9.9 2 7 0.0267 0.0309 Coarse 629.4 9.9 6 9 0.0282 0.0300 Coarse 629.4 9.9 6 4 0.0362 0.0449 Coarse 629.4 9.9 24 6 0.0243 0.0325 Coarse 629.4 9.9 24 3 0.0313 0.0527 90 ------- TABLE A-3. HTDRAULIC CONDUCTIVITY ESTIMATES FOR UNSTEADY DRAINAGE FSOM PRXSATURATED SAND Type of Sand Model Length (ii.) Slope (I) Depth (in.) Hydraulic (in Bousslntsq Conductivity . /hr) HELP Fine 305.4 1.8 14 0.0150 0.0156 "ine 305.4 1.3 12 0.0111 0.0110 Fine 3C5.4 1.8 10 0.013"! 0.0191 Fine 305.4 1.8 9 0.0147 0.0148 Flea 629.4 2.1 24 0.0079 0.0116 Fine 629.4 2.1 10 0.0155 0.0161 Fink 629.4 2.1 6 0.0201 0.0201 Fine 629.4 2.1 4 0.0238 0.0245 Coarse 305.4 1C.0 10 0.0243 0.0244 Coarse 305.4 10.0 5 0.0319 0.0434 Coarse 629.4 2.3 13 0.0260 0.0257 Coarse 629.4 2.3 7 0.0446 0.0454 Coarse 629.4 1.3 6 0.0456 0.0451 Coarse 629.4 2,3 3 0.0621 0.0647 91 ------- APPENDIX 3 COMPARISON! BETVEIN BOUSSINESQ SOLUTION AND HELP LATERAL DRAINAGE EQUATION The HELP lateral drainage equation was developed from numerical solutions of the Boussinesq equation for saturated unconfined lateral flow through porous aedia under unsteady drainage conditions. Consequently, the tvo methods should predict the same drainage rate under the same unsteady drainage conditions when the same values art used for the design parameters. ihia appendix compares predictions by the tvo aethods using actual test data to deterxiae how well the HELP equation represents the Boussinesq solution. In Section 6 hydraulic conductivities predicted by the tvo methods were briefly compared. Since the drainage rate computation is directly propor- tional to the hydraulic conductivity value in both aethods, Che estimates of hydraulic conductivity using measured values of drainage rata, depth of satu- ration, drainage length, jlope end infiltration rate can be directly coapared to deteraine the agreement between the methodw. Comparisons are made la this appendix for steady-state drainage, unsteady drainage following rainfall and uncteady drainage from presaturated sand using the hydraulic conductivity values presented in Appendix A. Comparisons are aljo made to determine th* effect.1? of slope, drainage length, type of sand and depth of saturation on the agreement between the tvo equations. Steady-state drainage differs from unsteady drainage since recharge alters the profile of the depth of saturation as a function of distance from the drain. Consequently, the drainage rate at a given average depth of satu- ration la greater for steady-state drainage than for unsteady drainage whtn the profile is falling, and is less than when the profile is rising. In actual drainage tests the amount of difference is confounded by the difference in air entrapment between the tvo modes of drainage. Comparisons between pre- dicted hydraulic conductivity values for steady-state and unsteady drainage by the numerical solution of the Boussinesq equttiou and the HELP lateral drain- age equation are summarized in Table B-L where the ratio of hydraulic conduc- tivity values is presented. The ratio of hydraulic conductivity estimates for steady-state drainage to those for unsteady drainage was 0.614 using the Bous- sinesq solution and 0,831 using the HELP equation. Therefore, using a con- stant hydraulic conductivity value for all rypes of drainage would produce greater error with the Bouaslaeirq solution than with the HELP equation. These tvo regressions are presented in Figure B-l. A ratio of I.000 and aa inter- cept of 0.00C0 indicate that the data are identical. In all of th-s regres- sions presented in this appendix, the intercept was not significantly different from zero. Therefore, the regressions were run using an Intercept of 0.0000 to deteraine the ratios, K1/K2, presented in Tables B-l and B-2. 52 ------- TABLE 3-1. BTDRA'JIIC CONDUCT! 7ITY REGRESSION ANALYSIS St™MAKT 95* Confidence Limits Data Set Kl 'K2 K1/K2 Intercept s* £ r Steady-states drainage tests 1.416 1.15I to 1.680 -0.CC36 to 0.0035 7 0.992 All unsteady drainag- tests 1.134 1.052 to 1.217 -O.0C23 to G.0Q23 93 0.959 Unsteady drainage following rainfall 1.152 1.059 to 1.246 -0.0026 5o 0.0026 79 0.955 Unsteady drainage with presaturation 1.049 0.927 to 1.171 -0.0035 to 0.0038 14 0.991 Estimates from Boassines; solutiont 0.614 0.285 to 0.942 -0.00"6 to 0.0071 6 0.960 Estimates from HELP equationf 0.831 0.406 to 1.255 -0.010C to 0.0093 6 0.964 N • Number of values la data see. ** Kl - Hyditulic conductivity value using BELP equation. K2 ¦ Hydraulic conductivity value using Soussinesq solution, t K1 * Hydraulic conductivity value for steady-state draisage. K2 • Hydraulic conductivity value for unsteacy drainage. The 95-percent confidence llaits for the ratic and the intercept ara also presented in tables B-l and 8-2 to show the significance of the results. The ratio of the HELP te iate (KS) to the Soussinesq estimate (KB) for steady-state drainage is seae. -ically different at 95-percent percrat con- fidence from the ratio for unsteady drainage. The ratio for steady-state drainage vaa 1.416 while it was 1.134 for unsteady drainage. The rrgressions ere shown in Figure B-2. The difference in the ratios for unsteady drainage following rainfall and fron presaturated sands as shown in Figure B-3 was net significant at 95-percent confidence, but the ratio for presaturati^n was smaller and not significantly different from l.OCO. The HELP hydraulic conductivity values are significantly greater than the values estimated by the Soussinesq solution. As presented in Section 6, this means that the lateral drainage rate is underestimated bj the HELP model in comparison to the Boussioesq model by j»n average of 29 percent for steady-state draisage and 12 percent for unsteady drainage. Six par^eters w»re examined in the laboratory drair.age tests: type of sar.d, slope, depth of saturation, draisage length, rainfall duration and ------- 032 a HELP o BCUSSINESO HELP 0)6- BOUSSINESQ CD 008- .016 UNSTEADY K (cm/s-c) .024 .032 .008 Figure B-l. Comparisons of hydraulic conductivity values for ste£dy-state drainage to values for unsteady drainage as estimated by the HEI.P equation and the numerical Bousslnesq solution. ------- TABLE 1-2. HYDRAULIC CONDUCTIVITY REGRESSION ANALYSIS FOR UNSTEADY DRAINAGE FOLLOWING SAINTALL SIX Confi dence Lir.its 2 r Data Set KB/ICS KH 'KB Intercept N** Fine Sand 1.053 0.99* to 1.112 -0.0C56 to 0.0079 42 0.993 Coarse Sand 1.162 0.930 to 1.395 -0.0082 to 0.0096 37 0.953 Short Model 1.157 1.018 to 1.295 -0.0045 to 0.0036 43 0.950 Long Model 1.145 1.023 to 1.266 -0.0025 to 0.0037 36 0,964 21 Slope 1.033 1.011 to 1.055 ~0.0007 to 0.0007 39 0.999 51 Slope 1.216 0.800 to 1.632 -0.0083 to 0.0112 12 0.940 102 Slope 1.427 1.235 to 1.619 -0.0063 to 0.0026 28 0.95? Average Depth of 0" to 7" 1.195 0.946 to 1.443 -0.0087 to 0.0113 27 0.951 Average Depth of 8" to 14" 1.039 0.935 to 1.143 -0.0018 to 0.0024 31 0.984 Average Depth of 15" to 20" i.022 C.937 to 1.106 -0.0007 to 0.0013 21 0.991 y - 0" to 7" 21 Slope 1.044 0.995 to 1.092 -0.0025 to 0.0023 11 0.999 y - 0" to 7" 102 Slope 1.503 1.016 to 1.989 -0.0164 to 0.0146 13 0.964 y - 8" to 14" 2Z Slope 1.003 0.964 to 1.042 -0.0008 to 0.0010 16 0,999 y - S" to 14" 102 Slope 1.136 0.355 to 1.416 -0,0039 to 0.0039 11 0.957 y - 15" to 20" 21 Slope 1.024 0.903 to 1.145 -0.0013 to 0.0022 12 0.992 y - 15" to 20" 102 Slope 0.935 0.664 to 1.206 -0.0016 to 0.0016 e 0.999 * ra « Hydraulic conductivity value using HELP equation. O • Hydraulic conductivity value using Bouesinesq solution. ** N - Number of "aluea in data sat. 95 ------- . 1 \£j OS .08- a a) (/> E ------- . 1-1 ~ FOLLOWING RAiNFALL o PRCSATURAT10N 08" o Q) (/) E O Q. LxJ 06- FOLLOWINQ RAINFALL PRESA TURA TION , ! .030 .045 .060 BOUSSJNESQ K (cm/sec) Figure B-3. Comparisons of hydraulic conductivity astlmates by tlie HELP equation to estimates by the numerical Bousslnesq solution for unsteady drainage following rainfall and ftoni presaturated sands. 075 ------- rainfall Intensity. Inspection of the data indicated that rainfall duration and intensity had no effect on the drainage rate for a given sand, slope, length and saturated depth as depicted by the hydraulic conductivity esti- mates. This was verified by performing an unequal two-way analysis of variance (ANOVA) using slope as the blocks and either rainfall duration cr ittensfty as.the treatments. Blocks are the different levels of a variable that produced variance in the measured result. The levels of slope for the blocks vere 2, 5 and 10 percent. Treatments are the different levels of the second variable that produce variance (6). S.trpe was selected for the blocks siince it shewed the greatest effects on the ratios of drainage rates presented in Section 6 and the ratios of hydraulic conductivity valt es as shown in Table B—2 . The effects of the remaining four paranetera on the ratio of \ie hydrau- lic conductivity estimates were examined by two methods. The first method was linear regression of the HHP-generated and Boussinesq-generated estimates of hydraulic conductivity. This method does not examine interaction effects and can yield erroneous results due to interactions and cross-correlations. Therefore, unequal two- and three-way ANOVAs on the 79 ratios of the hydraulic conductivity estimates for unsteady drainage following rainfall were used as the second rethed because of their ability Co accoun. for interactions between variables. The results of the linear regression analyses for '.ype of sand, drainage length, slope, average depth of saturation and average depth of saturation at a given slope are presented in Table B-2. The ratio for estimates generated with data from drainage tests using fine sand was 1.0S3, while it was 1.162 for estimates for coarse sand. The regressions are shown in Figure 1-4. How- ever, the confidence interval for the coarse sand ratio was large and included the confidence limits for the fine sand ratio so the type of sand did not sig- nificantly affect the ratio. Eowever, since the size of the confidence inter- vals differed by a factor of 3, the type of sand may have an interaction with at least one of the other variables. The ratios for short model estimates and long model estimates are very Jiailar as shown in Figure 1-3 and therefore the drainage length does not significantly affect the estimates. The ratio of KH/O increased significantly with increasing slope as shown in Figure B-6, lr-dicatlng chat the HELP model underestimates the effects of slope in increas- ing the drainage rate as predicted by the Bcussinesq solution. The effect of depth of saturation is shown In Figure 3-7 where the hydraulic conductivity estimates were divided into three groups based on the depth of saturation for which the estimate was generated. These ratios were not significantly differ- ent because the confidence Interval for tha ratio at depths of saturation ranging from 0 to 7 inches wra such larger than the others, As with the type of sand, an interaction may he occurring with another variable. The interaction of slop* and depth of saturation wa® examined by perform- ing separate regressions for the effect of depth of saturation using estimates of hydraulic conductivity for drainage tests at 2-percent and 10-percent slopes. Figures B-8 and B-? show the regressions for 2-percent and 10-percent slopes, respectively. At 2-percent slope the ratioi did not differ at the three ranges of depth of saturation. At 10-percent slope KH/KS decreased sig- nificantly with increasing depth of saturation. Consequently, the MEL? 98 ------- O FINE SAND a COARSE SAND .08- u QJ " . Ofi- E U COARSE SAND FINE S.tND I LlJ m .02- .030 .060 .015 .075 NOUSSINESQ K (cm/sec) Figure B-A. Comparisons of hydraulic conductivity estimates by the HELP equation to estimates by the numerical Bousslnesq solution for unsteady drainage from fine and coarse Bands. ------- O SHORT MODEL ~ LONG MODEL .00" o o (J Q) \ .06- E (\ SHORT MOOEL LONG MODEL Q_ UJ ~C .045 (cm/ .030 BOUSSINESQ K (cm/sec) .015 Figure B-5. Comparisons of hydraulic conductivity estimates by the HF.l.P equation to estimates by the numerical Rousslnesq solution for unsteady drainage from the short and long physical models. ------- Q 27. SLOPE o 5Z SLOPE * m SLOPE . 00H /-N u 01 \ .06- E a 10% SLOPE oO " 2'/. SLOPE UJ :c .02- o- .060 .075 .015 .030 B0USS1NESQ K (cm/sec) m/sec) Figure B-6. Comparisons of hydraulic conductivity estimates by the IIEI.P equation to estimates by the numerical Bousslnesq solution for unsteady drainage froia models at 2-, 5- and 10-percent slopes. ------- TO 20 r •• .08- u Ol \ .06- E U TO 7 .04- Q_ 111 X .02- . 060 . 075 .015 . 030 .045 DOUSSINFSCI K (cm/sec) Figure B-7. Comportuona of hyiir/vjllc conductivity estimates by the HEI.P equation to estimates by the numerical RouBslnesq solution for unBtendy drainage ««t depths of saturation ranging from 0 to 7 in., 8 to 14 In., and 15 to 20 In. ------- .08 0" 10 .06- 9=0' TO 7r ^ o OJ U) E a ~ .04" TO 14 15' TO 20 Q. la I .02- .0)5 .030 DOUSSINESQ K (cm/sec) .060 .045 (cm/se c Figure B-fl. Comparisons of hydraulic conductivity estimated hy the HELP equation to estimates by" the numerical Bousslnesq solution for unoteady dralnag'. from models at 2-pt.-cert slope wlfh depths of saturation ranging from 0 to 7 In., 8 to 14 in., and 15 to 20 In. ------- . 1 o i- 0 01 <7> N & U a. U) .00- 06- . oH .02- ° y o y A y 0" TO 7" 8" TO 14" 15" TO 20" y-16 TO 20 1 .020 BOUSSINESQ K .030 (cm/sec) Figure B-9. Comparisons of hydraulic conductivity estimates Hy the HELP equation to astlmatie by the numerical Bo'iselnesq solution for unsteady drainage from modelt* at 10-percent slope with depths of saturation ranging from 0 to 7 in-, H to !4 In., and 15 to 20 In. ------- equation underestimated lateral drainage at small depths and large slopes compared to the Boussinesq solution but agreed well at depths above 14 in. Additional comparisons of the effects of design parameters and their interactions on K3/KB were made using unequal thr?e-vay ANOVA. Use ;f ANOVA on the ?9 ratios for unsteady drainage relieving rainfall indicated that the main effects of slope, depth of saturation and drainage length and the interactive effects of slope with drainage length and slope with depth (listed In their order of importance) were all significant at 95-percent confidence. tf3iag AHOVA on the 42 ratios froo the tests on fine sand, only the effect of slope was significant at 95-percent confidence. For the 37 values from tests an coarse sand, the interactive effect of slope with drainage length and the mtfin effect of slope vera significant at 95-percent confidence. Results of ANOVAs on the data sets divided by type of sand Indicates that interaction occurred between the type of sand and the other effects such as death of saturation, drainage length, and slope with depth, since these vari- ables were not significant upen dividing, the data set3. Therefore, an addi- tional unequal three-way ANOVA was performed examining the effects of type of sand, slope and drainage length using all 79 ratios for unsteady drainage following rainfall. Depth of saturation was replaced by type of sand in the analysis since depth of saturation was not significant in either of the two ATOVAs performed on data for one type of sand. T>e results indicate that the interactive effect of slope with drainage length and the main effects of slope, type of sand and drainage length (listeJ in the order of importance) are significant at 95-p«reent coofidince. In cone.1 usion, the HELP equation underpredicts the lateral drainage rate in comparison to the numerical solution of the Soussineaq equation by about 12 percent for unsteady drainage and by about 29 percent for steady-state drainage. The actual underpredicticn Is a function of slope, product of the slope and drainage length, type of sand, and drainage length. The tvo models treat these effects in significantly different ways but, nevertheless, fre- quently produce similar drainage rates. Under the worst conditions, the drainage rate was unierprediced by 50 percent and overpredicted by 50 percent during unsteady drainage. Under steady-state conditions, drainage was under- predicted by a range of 13 to 40 pe~cent. 105 ------- appendix c COMPARISONS BE7VETM LASORATORY KIASURJ^ENTS AND NUMERICAL 3CUSSI.VESQ SOLUTIONS The HELP lateral drainage equation was developed to approximate numerical solutions of the one-dimensional Boussinesq equation for unsteady, unconfined, saturated flow through porous media. Appendix B and parts of Section 6 of this report present comparisons between che ETLP equation and the Boussinesq solution to evaluate the validity of the EEL? equation. Therefore, it is necessary to compare the Boussinesq solutions to the laboratory measurements. This forss a basis for judging the significance cf the differences between the HELP equation predictions and the laboratory aeasureaents, and between che HELP equatien and the Boussinesq solution. This appendix presents the com- parisons and discusses briefly how well the Boussinesq solution predicts the drainage results. In addition, the performance of the HELP equation is com- pared to the Boussinesq solution results to present the significance of the differences between the HELP predictions and the laboratory aeasureaents. Figures C-l and C-2 show the range of measured drainage rat«s aa a func- tion of average head above eh* liner (average depth of saturation). Three predictions by the numerical Boussinesq solution using three hydraulic conduc- tivity conditions are shown for each test condition. The first curve is based on the hydraulic conductivity measured in the laboratory permeameters. The second curve was obtained by using an average of Che various hydraulic conduc- tivity values estimated for the test condition by attempts to calibrate the Boussio«sq solution. For the third curve, che hydraulic conductivity was entered as a pow»r function of the average depth of saturation. This equation vas developed by performing a curve fit of the hydraulic conductivity values obtained by calibration. Thus# power functions are presented in Section 6. These figures are analogous to Figures 25 and 26 in Section 6, where the predictions were obtained using the HELP model instead of the Boussinesq solution. The predictions using the lasoratcry-measured hydraulic conductivity value differ greatly from the actual results in all but one of the eight test conditions. Predictions based on the mean hydraulic conductivity values cali- brated for the test conditions performed better and generally showed agreement over a narrow range of average depths. Better agreement was obtained using che function for hydraulic conductivity, but the relationship did not fall within the range of observations throughout the entire range of average satu- rated depth despite all of the attempts to calibrate the Boussinesq solution. Comparing the results shown in Figures C-l and C-2 with Figures 25 and 26 shows that tne results obtained using the HELP equation with the same 106 ------- < £T •MOAT MOOtL •« tLOft AVlflAGt HtAU (inches) Prdict4 RmuII* * l«6 I o Hot M < z S T, I." " 1 AVfRACC lt£A(l (inchus) FrJut«4 IimIIii lot R Ui ~— ¦< oc •MO«T MOOIl f« HOM fT I. ! AVtRACC hfAO (inchos) n Fradidd KnuIUi • Id ¦ J. u »— oc X < a AVJRAbf MAD (incltus* Figure C-l. Measured drainage rate va. average head compared to numerical Bousslneaq solutions for fine sand In both physical models at elopes of 2 and 10 percent. ------- tONO UOOli am lion UJ »-• •< .15- QC z ¦< E Pr«dicl«d flodillii - t<4> « o Nao) K is ir~ AVCKACf HAD (inches) • HORf UOOIl • ft UOPI i m Ui »— * -< a Prdicl«d Rtiullii lot H O »«- • f (y) AVCkACf I1CAO (inchus) rnOAV MOOCL 10ft UOPk .16- Ui - ¦< ¦< Aridictld RainUii «ldiK .01- AVtRAW IK AO (inclias) Figure C-2. Measured drainage rate vs. average head compared to numerical Rousslnesq solutions for coarre sand If. both physical models at slopes of 2 and 10 percent. ------- hydraulic conductivity values produced results that were slightly better, though perhaps not significantly, than the Boussinesq solution, Thti result is surprising since the hydraulic conductivity values wete calibrated for the Boussinesq solution and not the HELP equation. Consequently, it is apparent that the HELP model performs as well as the Boussinesq solution. In addition, the differences between the two methods are much smaller than the differences between the predicted and actual results, and in some cases are smaller than the range in the actual results. Figures C-3 through C-6 show the measured drainage rate and the predicted drainage rate as a function of time using the Boussinesq solution with the power function of depth of saturation used for the hydraulic conductivity value. Figures C-3 through C-6 also show the measured average depth of saturation and the predicted depth as a function of time. These figures are analogous to Figures 22 through 30 in Section 6 wher- the predicted values were obtained using the HELP equation. The results shown in these figures vary greatly; some show good agreement, some show poor agreement, and some show good agreement only after infiltration cesses. The same is tru for both the drainage rate and the average depth of saturation. In general, the results were very comparable to the results obtained using the XIL? model but were significantly better fcr test conditions having 10-percent slope. The HELP model yielded better results for test conditions having 2-perttnt slope. The HELP equation predicted smaller drainage rates for a given average depth of saturation. Figures C-7 through C-10 show the measured and predicted head above "he dra ------- .11 24. JUIM Af 0.14 IN./ 9CRT NODa Af 21 SLOPE 11- - PREOICHO a ACTUAL I z .3 i.o-m rain a? a37 m/m sent MQOa Af IQX SLUPC .1- .ii- fcf 2 X I e.(H RAIN Af a 57 IM./M) SHORT NOOO. AT IQ1 SLOPt — PWOICTIO O ACTUAL II- • - - 14. I-M) RAIN Al 0. !4 I* /H SHORT NOOCl AT 21 SLOP! — WCOICTED o ACTUAL II- < TINE (Nmr«) TIME (hart) Figure C-3. Measured drainage rate and average saturated depth vr. time compared to numerical Bousslnesq solutions for tine cand In short model at aiopee of 2 and 10 percent. ------- ft 2-m KAIN A! 0 U IK IOC MOQII AT 101 SI.OPE H OBOO %.&** ^AIN Al 0.54 1M. /Hi IQrC illI Al Zt StlFi I UJ i I ioA llMt (hour) FINE (haurc) a B it >« mi r a st in /» limc Mm «i ;i aort - W?0ICUB a ACTUM. To ~ la riHf Ihotrt! fc 106 £ BM« ff&Jti A! a 54 IN. /m LONG MOOa Af 101 SLOP! -- PftfOfCIUi a ACltttt. 1—Ti—R "Ttt JT~G 17 I INC &««•»> «4 Kijjure C-4. Measured drafnagu rate and average s.imraied deptli vs. time compared to numerical Hhusn ltiea«\| solutions fur fine Hand in J model ill .slu\|iii, ;if 2 and in pcrcint. ------- L (M« IUIN AT (L SO IN. /Ml MM MOO. AT 21 SLOPE .01- o o n-m iuin ai a» in./>« SHOD I mXXL AI 102 SLOPE — pf&mcTco O ACTUAL X .1- a a a LO-t HA I.N AT 0. St IN. AA SMMI MOO. AT 21 SLOPC a- - PREDICTED O ACTUAL IS- M- O a T IMC (hours) L 1-tA IUIN AT a 56 IN. AH SHORT MlXl AI 102 9 OPE tt- e & i- f IMC (hours) Figure C-5. Measured drainage rate and average saturated depth va. time compared to numerical Bousslneaq solutions for coarse sand In short model at slopes of 2 and 10 percent. ------- «. i-mi hasm At a ¦M in /tit IK MGDU A! 21 &D* 0 o a o I nj $ TT r SMBE ftiourc) e z-m uin m o. 57 in. /m lflN& MUL 1 I (LI iiQPl PRCIICTCO a ACfUAL 1 U G—S—5 £—k 1 IMC Jhour> C l-W RAIN A? ftSfl IM. IOC MDa AT 21 RCPt pmiciro o ACTUM. 8. M9I RAIN Al 0 17 IM. /M iwt tmx a! iqc acre pwnicicD « actual illSO T IMC (ho,»«> lb Jo E lu Is IM (har> Menstired ilroJrate dinl average satur^t^d depth vs. t Jim: i on»\|>«.roJ ldil sl ------- • h ¦ 17.78* nil. PREDICTED « K ¦ 12.63* 05IN. ACrjM. c fi • ;7.6r FILL. ACUA1 LH UJ X k-t z 1% SLOPE 15- UJ > JO- CD 05 < O ^rr* °f * DISTANCE I'RCM DRAIN (FEET) * S * 37.03- ORAiN. PREDICTED • B • 43.81* PILL PREDICTED 44. 57- ? ILL ACTUAL uj X 10* MUOPt JZ a ns <« 20-1 9 LU HH DISTANCE FROM drain (FE£T) Figure C-7. Measured head profiles for unsteady drainage during 24-hr rainfall test at 2-percant slope and during 6-hr rainfall test at 10-percent slop* compared to numerical Boussinesc, solutions for fine sand in short model. 1U ------- 40- 15- o s • ;i50* ^?a:s. actui c h ¦ 21,79* nu_ ACTUA^ z 5» a cl -c a u_ ?C,0 30.0 DISTANCE FROM QRAiN (FEET) 40,0 SC. 0 50.3 CFE T Vi- sa «c a < • R • H.63* \UA1N. PRO ICED ¦ h • 47.56* Flu. W£CIC£D » h • 4 D h • 4 mo ZC. 0 3C.0 DISTANCE. FROM DRAIN 40.3 (FEET) 53.0 60. 3 Figure C-8. Measured he'1 profiles for unsteady drainage durinK 6-hr rainfall test craj.artd ro numerical Bousslne?<[ solutions for fine sand in long model &c slopes of 2 and 10 percent. 115 ------- * h . ;9.41* DRAIN, PREDICT^] ¦ h • 10.72* f:ii. WED!CE • 5 ¦ IKS* CHAIN, AC'Wl £ 25- x JZ z 15- «< QC cz UJ >- O 03 «C ItH o 5' ——— s' DISTANCE FROM DRAIN (FEET) 40 %r> Lj-J 2 z X z as a o m < o «e >* Figure C-9. • h • 25.88* DRAIN, BREEICTiJ F • 19. «* HLL PraiCTED • R • 25. 88* CfUJN. ktl-JL a R • 19.37* PIlL ACTUAL I0« StOrt T DISTANCE FRCM DRAIN iTEET) Measured head profiles for unsteady drainage during 6-hr rain- fall test compared to numerical Bouaainesq solutions for coarae sand ia short model at slopes of 2 and 10 percent. 116 ------- > • fi • i2. ;2" wain. f?n:ro • R • 20. 88" F!lI. DIETED • fi • 12. 13" ORAINITIAL Oh- 2C. Si'JiiL' ACTUAL Cn yj x LJ Z 20- gj > o CD «e id- distance FROM ZRA1N (FEET) tn 50- o m ¦< Q C uj X R • 4S.7T DRAIK WCTICTED « R • 38.9S# FILL TO3ICTE2 * R * 45. 6fl* VtAlK ACrjAL q fi • m f?- Pin, actual store • 10.0 2G.0 3D. 0 4£LQ DISTANCE FflOH DRAIN (FEET) 6H0 Figure C-10. Measured head profiles for unsteady drainage during 6-hr rainfall test compared to numerical Bousalnesq solutions for coarse sand In long model at slopes cf 2 and 10 percent* 117 ------- TECHNICAL REPORT DATA Ifitait rtsdInsinitttont or. the ttunt bt.arr eomrtttnnt! 1. PISCO*. NO. ZPA/GO.),'2-87/0^9 2. 3. ACCOUNT'S ACCESSION NO. PB SI- A 17 / OH- «. TlTi £ AND SuatiTli Verification cf the Lateral Drainage Component o' the HELP Model Using Physical Models S. «6fOT DATS Julv 19S7 . »0«MXC C AC AN E 2 A riON CSBi 7. uTKOiSI P.R. Schroeder and R. L. Peyton # OASAIltZAYlON NO 9 PtftFCRMINS ORC4N;2a:i0n n0. '«• kLIMINT MO. 11. CCNTAA:.T.<5fcANT f»0. DW-96930. ".6 13. 5»onSOBinO AGENCY nam£ AND AOOAESS Hazardous Waste Engineering Research Laboratory Office of Research and Development U.S Environmental Protection Agency Lmrinnat.i . Ohio 452fi£ 53. lypt Of £Por A'.o »€AiOC COVSAtB 14. JPONSOAINO A&iNC COOI EPA/600/12 is. sjCPlCmCsTaBy nOTIS Project Officer; Douglas C, Ammon 16. A»ST*CT This report describes a study conducted to verify the lateral drainage component of the Hydrologic Evaluation of Landfill Performance (HELP) computer model using laboratory drainage data from two la-ge-scale physical models of landfill liner/drainage systems. Drainage tests were run to examine the effects that drainage length, slope, hydraulic conductivity and depth of saturation have on the lateral drainage rate. The drainage results were compared with HELP model predictions 2nd numerical solutions of the Boussinesq equation for unsteady, unconfirmed flow through porous media. KIT WOK0S eOCUMlNT ANALYSIS J. DEISHIPT on fe.lOINTlFKMS/OFtN (NCIS TEAMS e. COSati Fitl-J/Grcup • :«. OlS''»"«WTIO STATEMENT Release to Public 'VmfYasYiWed 21. no. o« cages 1J 2 20. SICuhity class {T3mptl Unclassified 22. CiCi . EPA 2020-1 ((«». 4-771 «¦¦«<«u« «oition i «*l«k -------

EPA/600/2-87/0-19
July 1987
VERIFICATION OF THE LATERAL DRAINAGE
COMPONENT OF THE HELP MODEL USING PHYSICAL .MODELS
by
P. R. Schroeder and R, L, Peyton
L'.S. Artsy Engineer Waterways Experiaent Station
Vicksburg, MS 39180-0631
Interagency Agreement Number DV96930236-01-0
Project Officer
D. C. Aaanon
Landfill Pollution Control Division
Hazardous Waste Engineering Research Laboratory
Cincinnati, OH 4526a
HAZARDOUS WASTE ENGINEERING RESEARCH I .MORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OH 45268

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NOTICE
This document has been reviewed in accordance with
U.S. Environmental Protection Agency policy and
approved for publication. Mention of trade navies
or corn-rercij 1 products does r.ot constitute endorse-
ment or recommendation for use.
ii

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FOREWORD
Today's rapidly developing and changing technologies and Indue*. -Ill
products and prectices frequently carry with then the Increased generation af
solid and hazardous wastes. Th>:se materials, If improperly dealt with, car
threaten both public h*alth an*', the environment. Abandoned waste alt's anJ
accidetital rsleases of toxic and hazardous substances to the environaint also
have important environmental and public health implications. The Hazardous
Waste Engineering Research Laboratory assists in providing an authoritative
and defennible engineering basis for assessing and solving these proolemi.
Its products support the policies, progrsas and regulations of the environ-
mental Protection Agency, the permitting and other responsibilities of State
and local governments and the needs of both large and small businesses in
handling their wastes responsibly and economically.
This report describes a study conducted to verify the lateral drainage
component of the Hydrologic Evaluation of Landfill Performance (HEU'i comt.uter
¦odcl using laboratory drainage data from two large-scale physical Mdeli of
landfill liner/drain aystees. Drainage teatr were run to ixislat toe ef:ects
that drainage length, slope, hydraulic conductivity and depth of saturation
have on the lateral drainage rate. The drainage results were compared with
HELP model predictions and numerical solutions of the Boussinesq equation for
unsteady, unconfirmed flow through porous media. Detailed water budge*.s, test
liumarlis, and tables of the test data collected are published in appendices
ujder a separate report covi»r.
Thomas R. Hauser» Director
Hazardous Waste Engineering Researc.i Laboratory
Hi

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ABSTRACT
Two large-scale physical models of land±ill liner/drain systems ver*
consti-ucted to examine the effects that the length, slope and saturated
hydraulic conductivity cf the drain layer, and the u«>

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CONTENTS
Foreword				iii
Abstract						lv
Figures . 	 .....	vl
Tables 					xi
Acknowledgments 					xiii
1.	Executive Summary				1
Purpose and scope				1
Results of drainage tests				2
Verification of the HELP model		4
Conclusions and recommendations 	 ...	6
2.	Introduction 	 . 	 ........	7
Background ....... 	 . 		7
Description of HELP model 	 .......	9
Purpose ar.d scope 					13
3.	Model Design, Preparation, and Instrumentation 	 . .	14
Model design and construction		14
Test preparation		 ,	18
Instrumentation .......... 		21
4.	Experic«?nral Design and Procedures 				27
Experimental design 		27
Test procedure and data collection				30
Data reduction ....... 		32
5.	Physical Mcdtl kesults ... 	 ...........	33
Saturated-depth profiles 	 . 		33
Drainage rate and saturated depth		 .	33
Drainable porority 				52
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7IGVRES
Sur.be r	Page
1	Plan view of experimental components of the
physical aodels		 .	1
2	Cutavay s4 de view of physical xcdel			16
3	Cu:avay end view of phvsicai cedel at the trough:	17
4	Compact ion curves for buckshot clay sell	20
5	Grafn-size distributions of the sand drainage aedia 		22
6	Sketch of piezoneter atid transducer installation 		23
? Saturated depth profiles for unsteady drainage during 24-hr
ralr.fi.ll test at 2-percer.t slope and during 6-hr rainfall test
at 10-percent slope using fine sand in short raodel	34
8	Saturated deK th profiles for unsteady drainage during
6-hr rainfall rest using fine sar.d In long model
a; '.lopes cf 2 and 10 percent	35
9	Saturated deoth prof ilea for unsteady drainage during
6-hr rainfall test using coarse sand In short sodel
at slopes of 2 and 10 percent . 				36
10	Saturated dept! profiles for unsteady drainage during
6-hr rainfall test using coarse sand in Ion® model
at slopes ;-£ J and 10 percent		 37
11	Saturated depth profiles for steady-state drainage
in both physical models		 		38
12	Head profiles for unsteady drainage during 24-hr
rainfall test at 2-percent slope and during 6-hr rainfall
test at 10-percent slope using fine sand ir. short xodel ..... 39
13	Head prcfiles for unsteady drainage during 6-hr
rainfall test using fine sand in lon^ aodel
at slopes of 2 ard iO percent	40
vi
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Number	Page
14	Head profiles for unsteady drainage during 6-hr
rainfall test using r.o&rse sand In short model
at slopes of 2 and 10 percent			41
15	Head profiles for unsteady drainage during 6-hr
rainfall test using coarse sand In long acdel
at slopes of 2 and 10 percent	42
16	Head profiles for steady-state drain.iga in both
physical models 	 *3
\7 Average saturated depth and drainage rate as a
function of tiae for 24-hr rainfall test using
fine sand and for 6-hr rainfall test using
coarse sand in short model at 2-percent slope . 	 44
18	Average saturated depth and drainage rate as a
fur.ction of tiae. for 6-hr rainfall test using
both sands in short model at 5-percent slope 	 45
19	Average saturated depth and drainage rate as a
function of tiae for 6-hr rainfall rest using
both sjo43 in short model at 10-per-enc slope 	 46
20	Average saturated depth and drainage rate as a
function of tiae for 6-hr rainfall test using
both sands in long oodel at 2-percent slope 	47
21	Average saturated depth ar.d drainage rate as a
function of time fox 6-hr raiufall test using
both sands in long model at 5-percent slope 	48
22	Averse saturated depth and drainage rate as a
function of tiae for 6-hr rainfall test using
bsth sands in long model at 10-p«;rcent slope ..... 	 49
23	Unsteady drainage rate as a function of average
saturated defth for fine srnd in both physical
models 			 ...... 50
24	Unsteady drainage rate as a function of average
saturated depth for coarse sand in both physical
models 				51
25	Measured drainage rate followng rainfall vs.
average head above the liner compared to HELP
drainage equation predictions for fine sand in
both physical models at slopes of 2 and 10 percent ....... 62
vli
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Number
Page
26 Measured drainage rate following rainfall vs. average
head above the liner compared to HEL? drainage
equation predictions for coarse sand in both physical
models at slopes of 2 and 10 percent				 63
17 Measured drainage rate and average saturated depth vs.
time cospared to HELP prediction for fine sand in
the shoit model at slopes of 2 and lt> percent ... 	 . 65
28	Measured drainage rate and average saturated depth «s,
ties compared to HELP prediction for fine sand in
the long model at slopes of 2 and 10 per-ent	66
29	Measured drainage rate and average saturated depth vs.
tiue compared to HELP prediction for coarse sand in
the short model at slopes of 2 and 10 percent ......... 6?
30	Measured drainage rate and average saturated depth vs.
tiee ccmpa^d to HELP prediction for coarse sand in
the long model at slopes of 2 and 10 percent 		68
31	Measured drainage rate and average saturated depth vs.
tine cospared to HELP prediction for fine sand in
the long model at 10-percent slope using mean values for
hydraulic conductivity and dralnable porosity	• 69
32	Measured drainage rate and average saturated depth vs.
time compared to HEL? prediction for coarse sand in
the «hort oodel at 10-percent slope using mean values for
hydraulic conductivity and dr»lr>sble porosity		 70
33	Predicted »s, actual drainage rate using measured drainage
from short model to predict drainage from long model ...... 7j.
34	Predicted vs. actual drainage rats using measured drainage
at 2-percent slope to predict drainage at 10-p*rcent slope ... 76
35	Predicted vs. actual drainage rate using measured drainage
at y - 6 in. to predict drainage at y • 12 in. ........ 78
36	Predicted vs. actual drainage rates using measured drainage
at al • 15.3 in. to predict drainage at aL -
30.5 in					80
37	Predicted vs. actual drainage rate using measured drainage
from short model to predict draincge from long model
having the same value of aL ...... 	 ....... 82
B-l Comparisons of hydraulic conductivity values for steady-state
drainage to values for unsteady drainage aa estimated by
the HELP equation and the numerical Boussinesq solution .... 94
viii
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Mumber	Page
9-2 Comparisons of hydraulic conductivity estieates by the
HFL? equation to est isa-tes by the numerical Boussir.esq
solution for steadv-state and unsteady drainage 	 96
3-3 Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Boussinesq
solution for unsteady drainage following rainfall and
frota presaturated sands		 		 97
B-4 Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Boussinesq
solution for unsteady drainage froa fine and coarse sands ... 99
8-5 Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Boussinesq
solution for unsteady drainage from short and lor.g
physical models 	 100
B-6 Comparisons of hydraulic conductivity estimates by the
HFL? equation to estiaates by the numerical Boussinesq
solution for unsteady draicagt from models at 2-, 5-
and 10-percent slopes			 . 101
B-7 Comparisons of hydraulic conductivity estimates by the
KELP equation to estiaates by the numerical Boussir.esq
solution for unsteady drainage ct depths of saturation
ranging from 0 to 7 in., 8 to 14 in., and 15 to 20 in	102
B-8 Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Boussinesq
solution for unsteady drainage from models at 2-percent
slot^e with depths of saturation tanging trom 0 to 7 in.,
8 to 14 in., and 15 co 20 in			
2-9 Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Boussinesq
solution for unsteady drainage from models at 10-percent
slope with depths of saturation ranging from 0 to 7 in.,
8 to 14 in., and 15 to 20 in. . 			 .
C-l Measured drain*;* rate vs. average head compared to numerical
Boussinesq solutions for fine sand in both physical models
at slopes of ? and 10 percent 			 .
C-2 Measured drainage rate vs. average head compared to numerical
Boussinesq solutions for coarse sand in both physical models
at slopes of 2 and 10 percent		108
C-3 Measured drainage rot* and average saturated depth vs. tine
compared to numerical Boussinesq solutions for fine sand
in short model at slopes of 2 and 10 percent	110
. . 103
. . 104
. . 107
lx
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Number
Page
C-4 Measured dtair.age rate and average saturated depth vs. tias
compared to numerical 3ouss"!nesq solutions for fine sand
In long sodel at sio?«3 of 2 and 10 percent	Ill
C-5 Measured drainage rate and average saturated depth vs time
compared to numerical Boussinesq solutions for coarse
sar.d in short mode] at slopes of 2 and 10 per cent	112
0-6 Measured drainage rate and average saturated depth vs. tine
compared to numerical Boussinesq solutions fcr coarse
sand in Isng socle 1 at slopes of 2 and 10 percent	113
C-? Measured head profiler for unsteady drainage during 24-hr
rainfall test at 2-percent slope and during 6-hr
rainfall test at 10-percent slope compared to numerical
Boussinesq solutions for fine sand ir. short model	114
C-8 Measured head profiles for unsteady drainage during 6-hr
rainfall test compared to numerical Boussinesq
solutions for fine sand in long model at slopes of
2 and 10 percent	115
C-9 Measured head profiles for unsteady drainage during 6-hr
rainfall test rompared to numerical Boussinesq
solutions for coarse sand In short model at slopes of
2 and 10 percent			116
C-IQ Measured head profiles for unsteady drainage during 6-hr
rainfall test compared to numerical Bous-*ir.esq
solutions for coarse sand in l^ng model ct slopes of
2 and 10 percent	117
-------
T\BLES
Nusber	Page
i	¦ JB»	
1	Experimental Conditions £cr Unsteady Drainage Tests with
Rainfall			28
2	Experimental Conditions for Steady-State Drainage Tests 	 29
3	Experieental Conditions for Drainage Tests Usir.g Presaturated
Send			30
4	Drainable Porosity 				53
5	Values for Drainable Porosity Constant, DPC 	 54
6	Drainage Uses					55
7	Coaparison of Cocnutea Hydraulic Conductivities for
Steady-State Drainage 	 ... 57
8	Coaparison of Computed Hydraulic Conductivities far
Unsteady Drainage 				59
9	Regression Coefficients for Hydraulic Conductivity as a
Power Function of y	61
10	Effect of Length in the HELP Drainage Equation . 		72
11	Regression Analyses Sunitary for Evaluation of the HELP
Lateral Jrainage Equation 	 . 	 ..... ~4
12	Effect of Slope in the HELP Drainage Equation	75
13	Effect of Average Depth of Saturation in the HELP
Drainage Equation .... 	 77
14	iffect of al in the HELP Drainage Equation	79
15	Effect of Length in the KELP Drainage Equation Given
ronstant aL 		81
A-l Hydraulic Conductivity Esticates for Steady-State Drainage .... 87
A-2 Hydraulic Conductivity estimates for Unsteady Drainage
Following Rainfall ..... 	 86
xl
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Nuaber	Pagt
A-3 Hydraulic Conductivity Estimates for Unsteady Drainage
froo Presaturated Sand 		91
8-1 Hydraulic Conductivity Regression Analysis Sunmary 		93
B-2 Hydraulic Conductivity Regression Analysis for Unsteady "
Drainage Following fi3infall , . . . 				95
xil
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ACKNOWLEDGMENTS
Toe authors would like to express their sincere appreciation to
Mr. Sidne;* Sagsdale of the Environmental Engineering Division (EED), Environ-
mental L?joratory (EL), U.S. Army Engineer Vaterva s Experiment Station (WES)
and Kr. ?>oy Leach cf the Geotechnical Laboratory (GL), VES, for their help in
designing, constructing, and operating the physical models. The authors wish
to acknowledge Mr. Stafford Cooper, GL, for designing the jacks for the model;
the Engineering and Construction Services Division, VES, for constructing and
preparing the physical models fc* testing; and the Instrumentation Services
Division, WES, in particular Kr. Thomas McEven, for instrumenting the physical
ncdels. The authors also wish to acknowledge Mr. Thoias E. Schaefer, Jr., and
Mr. Roy Wade of the EED, VES, for running the drainage tests. The authors
would like to- thank Mr. Anthony Gibson and Mses. Charlotte Harness, Anita
Zltta and Kathy Smart of the £ZD, WES, for their many contributions of tech-
nical support leading to the completion of this study. In addition, the
authors would like to acknowledge the support and general supervision of
Mr. F, Douglas Shields and Dr. Raymond L, Montgomery of the EED, WES. The
authors would also like to thank Dr. Robert Havis ar.d Dr. Bruce McEnroe of the
EED, WES, for their techrical review and the Information Products Division,
WIS, for preparation of the final figures and editorial review.
xlii
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SECTION 1
EXECUTIVE SUMMARY
PURPOSE AND SCOPE
This study was conducted to test and verify the liqaid management tech-
nology for lateral subsurface drainage in covers and leachate collection
systems. The specific objective was to verify the lateral drainage component
of the Hydrologic Evaluation of Landfill Performance (HELP) Model (1,2) and
other regulatory and technical guidance, provisions and procedures developed
by the U.S. Environmental Protection Agency (JSEPA) (3),
The HELP model is a computer model that generates water budgets for a
landfill by performing a daily sequential simulation of water movement into,
through and out of the landfill. The model produces ear'.mazes of depths of
saturation and volumes of runoff, evapotranspiration, lateral drainage, and
percolation. Lateral drainage is computed in the model as a function of the
average depth of saturation above the liner, the 3lope of the surface of th#
lir.ir, the length to the drainage collector, and t"*.e hydraulic conductivity of
the lateral drainage layer (1). Therefore, to accomplish the objective of
this study, the lateral drainage rate was measured as a function of the
hydraulic conductivity, slope, lenrth and depth of saturation of the lateral
drainage layer in large-seal* physical models. The measured average depths of
saturation, drainage rates and drainage times in the physical models were then
compared with HELP model predictions ,md numerical solutions of ths Boussinesq
equation, which applies Darcy's lav t > unsteady, unconfirmed flow through
porous m»dici.
Two large-scale physical models of landfill liner/drain systems were con-
structed and filled with a 3-foot (iij sand drain l.yer overlying a 1-ft clay
liner, A 2-inch (In.) layer of gravel was placed under tHe liner to collect
seepage from the clay. The models were instrumented to measure the water
table profile, subsurface lateral drainage rate, water implication, runoff and
percolation through the liner. Evapotranspiration and other water losses were
estimated from the water budget for each test. Ths models have adjustable
slope, ranging from 2 to 10 percent in rhi3 study; and different lengths, one
being 26.5 ft and the other being 53,5 ft.
Several drainage tests were run on each configuration of the models by
applying water as rainfall to the surface ot the send layer, and then measur-
ing the water table along the length of the models and the lateral drainage
rite as a function of time. Lateral drainage rates arid wa'et table profiles
vere measured during periods of increasing, decreasing and steady-state drain-
age rates. In these drainage tests, two drainage lengths were
1
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compared'—25.4 ft and 52.4 ft. Three slopes were examined—approximately 2, 5
arid^lO percent. Sands of two hydraulic conductivities we je used—4 x
10 cent Ir.eter.'second (era/sec) (fine sand) and 2.2 x 10 ca/sec (coarse
sand) as measured In soil testing pemeaaetars. Four rainfall events were
examined—a 1-hour (hr) rainfall at 0.50 inches/hour (in./hi), a 2-hr rainfall
at 1.50 in./hr, a 6-hr rainfall at 0.50 in./hr and a 2i-hr rainfall at
0.125 lr,./hr. Also, water wcs applied to the sand for a long period of tLie
(generally mors than 36 hr) at a rainfall intensity which would maintain the
average depth of saturation In the sand at 12 in. In addition to thest
drainage tests, the sand was saturated, predominantly from tha bottom up for
several test conditions, and then allowed to drain. In total, mot« than sixty .*
tests were performed.
A complete block sxperimertal design was used to examine the effects of
drainage length, slope, hydxaulic conductivity, depth cf saturation, rainfall
intensity and rainfall duration on the lateral subsurface drainage rates. The
block design was selected because it provided the aoct data with the least
tine and expense for construction and model preparation. Several slopes and
rainfall events caulc, be examined quickly since very little else was tequirad
for changing these test conditions. Aiso, the time requirements and costs for
running an additional test with a different slope or rainfall were less thin
10 percent of the requirements for preparing the model for a different sard.
Additional rainfall events were examined in lieu of replicates since the
lateral drainage rate as computed by the HELP model does not directly cor.slder
the effscts of rainfall Intensity or duration. Also, since a complete block
design was used, the effect of a change in a variable is directly examined
under oultiple test conditions, reducing the need for replicates.
RESULTS OF DRAINAGE TESTS
A comparison of profile shapes for the depth of saturation alcn-£ the
length of the drainage layer indicates significant differences between the
rising saturated-depth profile (during filling) and the falling saturated-
depth profile (during draining) for the same average depth of saturation (y).
The prof litis are steeper naar the drain when filling than when draining. The
difference Is greater for higher infiltration rates. Steady-state profiles
are very similar to the profiles Cor draining.
The drainage rate for a given average depth of saturation was greater
during the filling portion of each experiment than during the draining
•portion. ThL% is consistent with the saturated-depth profiles whith show
steeper hydraulic gradients r.ear the drain for filling conditions. Plots of
drainage rates as a function of average depth of saturation also show that
draiazge continues aftsi.* y has essentially reached zero. This Is presuced to
be drainage of capillary water, commonly called delayed yield. An estimate of
this capillary water volume when y had just drained to 0 in. based on an
analysis of the experimental data Is about 0.1 in. (cubic inches per square
inch) for the fine sand and 0.3 is. for the coarse sand.
The drainag: results indicated that the drainable porosity of the sands
decreased with increasing depths of saturation above the clay liner. In
addition, the drainable porosity at all depths was considerably smallar than
2
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the value estimated from soli moisture dat? and other soil properties
collected on the sands. Low dralnable porosity vsiues were obtained In part
due to the delayed yield and capillary effects that results from the high
drainsge race. However, the the presence and vertical distribution of
entrapped air appear to be primarily responsible for the low dralnable
porosities and the change in dralnable porosity with height, although no
measurements of entrapped air were collected.
All parameters required to compute the drainage rate by the KELP equation
except the hydraulic conductivity were measured for each drainage test. Due
to variable air entrapment and differences in placement, compaction and
preparation of the sand drainage media, the hydraulic conductivity measured In
a permeaneter in the soils testing laboratory differed significantly from the
actual test values calculated from data on drainage rates and depths of
saturation from the physical models. As described in the documentation report
for the HELP model (1), the lateral drainage equation was developed to
approximate numerical solutions of the Bousslnesq equation fcr on«»-
diaensional, unsteady, unconfined flow through porous media. Therefore, the
actual hydraulic conductivity for che drainage tests was estimated by
adjusting ita value while solving the Bousslnesq equation until the results
matched the measured drainage r>ites and saturated depths. The hydraulic con-
ductivity estimates are summarized In Appendix A. Determining the hydraulic
conductivity in this taanrsr provided the best estimate obtainable for each
test since the Bousslnesq solution is the commonly accepted representation of
the accual drainage process. Comparisons were made for both steady-state
drainage during rainfall and unsteady drainage following cessation of
rainfall.
The computed hydraulic conductivity values differed significantly from
the treasured values. For steady-stace drainage frcr. the fins sand, t!-e
average computed value was only 8 percent greater than the measured value,
while for unsteady drainage the average computed value was about 150 percent
greater. For steady-state and unsteady drainage from the coarse sand, the
average computed value was respectively 92 and 84 percent less than the
measured value. For both sands, the average confuted hydraulic conductivity
for unsteady drainage was twice as large as the computed hydraulic con-
ductivity for steady-srate drainage.
In analyzing the computed hydraulic conductivity values, It was apparent
that hydiaulic conductivity decreased with Increasing y. This is consistent
with the earlier hypothesis that the volume of entrapped air increased with
increasing distance above the clay liner. A larger volume of entrapped air
decreases dralnable porosity ana cross-sectional flow-through area, thereby
decreasing hydraulic conductivity.
The computed hydraulic conductivity values varied considerably between
tests on the same sand, even In the same model without disturbing the
placement of the sand between tests. Considerable variability occurred
between tests having exactly the same configuration of sand, slope, Icr.gth,
and depth of saturation, where only the rainfall intensity and duration
differed. This variance was examined using an unequal three-way analysis of
3
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variance (ANC-VA) test to determine whether the computed hydraulic conductivity
•¦fas a function of another variable beside! average saturated depth.
The test variables used in the ASCVAs included type of sand, average
saturated depth, slope, drainage length, rainfall duration and rainfall
intensity. Ho effects of rainfall duratio-i and intensity could b<- discerned
by inspection; therefore, the initial AKOVAs were run using depth, slope and
length as the variables for data sets containing hydraulic cor.duc.ivity
•stiaates for one type of sand. These ANOVAs indicated that the computed
hydraulic conductivity estimates for both sands varied as a function of
average saturated depth and slope. Additional A>'GVAs indicated that drainage
length, rainfall intensity and duration did not significantly contribute to
thc variance in the computed hydraulic conductivity values. No physical
reasons are apparent for the variability of the hydraulic conductivity as a
function of slope. Therefore, the variability due to slope probably arises
from inaccuracies in the manner in which th# effects of slope are modeled by
the, boussinesq equation.
VERIFICATION OF THE HELP MODEL
The drainage rates computed by the. HELP t.odel was compared with the
results of the drainage tests in several manners. The hydraulic conductivity
that was needed to yield the measurer* drainage rate for the sa=r. drainage
length, slope, and aves.nge saturated depth existing in the drainage test was
computed for several times during each ti st. Thi 3 hydraulic conductivity
valje we9 cosrpa.?d with the vaJue measured in th* soils testing laboratory and
the "alue estimated using tha 3oi.asir.esq equation. If the HELP equation
accurately predicted the result? of the drainage ;ests, the hydraulic con-
ductivity value would agree witn the measured rr estimated hydraulic con-
ductivity value for the sand. If the h/drauliw conductivity value was greater
than t'ae value obtained for the sar.d, the HEL"1 equation underpredicted the
lateral drainage rate, Anorher method of comparison was to exanine the
effects of changing a single variable on the lateral drainage rate measured in
the drainage tests and predicted by the HELI* equation. The effects of
drainage length, slope, average depth of saturation, and the head contributed
by the slope of the liner were compared in this manner.
The hydraulic conductivity of the sand at various depths of saturation
was estimated for eacn test using thi Boussinesq solution of Davcy's law for
unsteady, unconfined flow thrcugh parous media and the HELP latiral drainage
equation. These hydraulic conductivity values were coopered to deteraine the
agreement between the KELP model and the Boussinesq solution, for steady-
state drainage, tne HELP model istiaates of th* hydraulic conductivity were
44 percer.t greater than the Boui sine?4 solution estimates. This result me&r.s
that the HELP model underestine ed the steady-state lateral drainage rate
predicted by the Boussinesq solut .on by 30 percsnt. The HELP estimates were
31 percent greater than the laboratory measurements for the f'.ne wind and
88 percent less than the laboratory measurements for the coarse s tnd. For
unsteady drainage, the HELP model estimates were only 13 percent greater than
the Boussinesq solution estimates which would underpmdict the lateral
dnirage rate by 11 percent, Th« closeness of the estimates was not unexpected
since the HELP lateral drainage equation was developed from numerical
4
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solutions of che Boussinesq equation for saturated unconflned l.-.teral flow
through poreus media under unsteady drainage conditions. The t.nderprediet ion
of the cumulative lateral drainage volume would be expeced te be very snail
since the removal rate of wai:er frosc the drain layer by ail other means is
much smaller than the lateral drairage rate. Consequently, che effect of dif-
ferences in tha predicted and actual drainage times are sral!.
The differences between t'.se laboratory measurement of the hydraulic con-
ductivity and either of the two estimates computed from drainage data were
much larger Chan the differences between the estimates. The HELP model and
Boussinesq equation predicted very similar drainage rates at 2-percent slope
but the HELP model predicted lower drainage rates at 10-percent slope. Unlike
the laboratory measurements, the hydraulic conductivity in the drainage tests
.arled as a function of the depth of saturation apparently due to entrapment
of air in the sand. This phenomenon makes it very difficult to model the
lateral drainage process and produce good agreement betwe-so the predicted and
actual results for drainage rate ar.d depth of saturation as a fui.ction of
t late •
An analysis was perforaed to deternine how well the lateral drainage
equation in the HELP model accounts for the effects of drainage length, slope
of the liner, average depth of saturation and head above the driin contributed
by the liner in the estimation of the drainage rate. The drainage equation
overestimates the decrease in drainage rate resulting from an Increase in
length given the tame sand, slope, depth of saturation and head from the
liner. Using the dr*in*ge rate for a drainage length of 25.4 ft to predict
the rate for a length of 52.4 ft, the HELP model underpredicte3 che rate by
18 percent. The HELP equation overestimated the increase in drainage rate by
30 percent that resulted from an Increase in slope from 2 percent to 10 per-
cent. Similarly, the HELP equation overestimated the increase in drainage
rate by 2Q percent that resulted from increasing the height (head) of the
crest of the liner from 15.3 te 30,5 in. above the drain. The effects of
changes in the average saturated depth on the drainage rate ptedicted by the
HELP equation agreed very well with the actual results.
Since the HELP lateral drainage equation was developed t> approximate
uua-.wriCPi solutions of the one-dimensional Boussinesq equation for unsteady,
unconflned, saturated flow through -porous media, it cannot be expected to
perform any better than the Boussinesq equation. Therefore, ic was necessary
to compare the Boussinesq solutions to the laboratory measurements in order to
form a basis for judging the significance of the differences between che HELP
equation predictions ard the laboratory measurements, and between the HELP
equation and the Boussinesq solution.
To summarize, the Bouaslnesq solution after calibration still produced
results significantly different frc* those measured in the drainage tests.
The results obtained with che HELP sodel were generally as good or better than
the Boussinesq solution. The HELP equition performed better on teats con-
ducted with 2-pnrcent slope end the Boussinesq solution performed somewhat
better on teats conducted at 10-percent slope. The dlfferen:ea between
predictions by the two methods for a given set of conditions were small In
comparison to the range of actual results. Siallarly, the differences between
5
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the predictions and the actual results were much larger than the differences
between the HELp equation and the Bou«sInesq equation.
CONCLUSION'S AND RECOMMENDATIONS
The following conclusions and reccaaendations ars 3iade. Lateral drair.aps
in landfill lin<.-r/drain systems is quite variable, probably due to air entrap-
ment. The hydraulic conductivity measurement cade in the laboratory is qu'te
different than rhe ir.-place value. Consequently, the estieacior. of the l?t-
 determine whether the effects ;.re unique to this ''ata
set. Additional data should be collected for longer drainage lengths and
greater slopes and from actu il landfill/liner syster.s.
6
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SECTION 2
INTRODUCTION
BACKGROUND
Landfills have coat to be a widely employed means for disposal of munici-
pal, industrial and hazardous solid wasres. Storage of any waste m/.terlal In
a landfill poses several potential problems. Among these is the p-jsslble con-
tamination of ground and surface waters by th» migration of water or leachate
from the landfill to adjacent areas. Given this potential problea, it is
essential th3t the liquids oanageaent lechnclogy perform as expected over the
life of the landfill. It is also essential that the performance of the tech-
nology can be simulated or modeled with sufficient accuracy to design land-
fills to prevent aigra:ior. of liquids froc the facility. The modeling of the
moisture movement through landfills also provides important information for
review of landfill designs and evaluation of the adequacy of the design and
the limitations of the liquids management technology.
This study was conducted to test and verify the liquid management tech-
nology for lateral subsurface drainage in covers and leachate collection sys-
tems. The specific objective was to verify the lateral drainage component of
the Hydrologic Ev?luatior. of Landfill Performance (HELP) Model (1,2) and other
regulatory and technical guidance, provisions and procedures developed by the
U.S. Environmental Protection Agency (L'SEPA) (3).
The USEPA regulatory provisions for lateral drainage layers require only
that the depth of leachate buildup at the bottom oi the landfill should not
exceed I ft and the construction materials for i>.aerate collection should be
resistant to chemical attack and the physical forces exerted on them (3).
USEPA technical guidance states that the drainage layer should be constructed
to be at least 12 In. thick at a minimus slope <^f - percent %nd have a
hydraulic conductivity of not less than 1 x iO cm/sic. Also, the drainage
pipe system should be of appropriate size and spacing -o efficiently remove
the leachate. It is believed that 4-in,-diameter pipes spaced 50 tc 200 ft
apart would be adequate for removing leachate (3). The drainage layer in the
cover shculi have the same specifications as above except that pipe drainage
systems are not necessary, although free drainage must be provided at the
perimeter of the cover (3).
Subsurface drainage has been a subject of interest for at least 200 years
as san attempted to drain marshes and reclaim land for health and agricultural
purposes. The literature is filled with citations describing drawdown of a
water table under steady-state conditions with pipes placed la parallel at a
constant elevation. In the majority of these studies the impervious barrier
7
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soil layer was well below the elevation of the drain pipes. Drainage unier
these conditions has been fully described and can be predicted vithin
10-percent accuracy. The phreatic surface is elliptical as lon& as the drains
reaain unsubmerged and do not restrict drainage.
Drilnage from soils where the drainage pipes are plactd on the surface of
the barrier soil layer is less well defined, in general, the drainage for
this condition is considerably slower, and the accuracy of the drainage rate
estimate by Dupuit's law (Darcy's law for unccnfined flow) is slightly worse,
though reasonably good. The drainage rate Is smaller because the cross-
sectional area through which the water flows toward the drain is sraaller. The
drainage rate is slightly overestimated because the flow is more curvilinear,
violating the parallel flow assui.ption.
Studies on subsurface drainage from soils above a sloped Impervious layer
are not widely found in the literature. Preliminary findings on drainage
through pipes placed well above the barrier soil layer have been repo-ed, but
none of these studies examined drainage from the surface of the barrier soil
layer as performed in landfills and in this study. The literature does not
present equations to predict the drainage rate or the phreatic surface.
The majority of the drainage equations reported in tha literature were
developed using, 3 steady-state assumption of ? uniform recharge rate equal to
the drainage rete. The drainage equation used in the HELP model assumed
steady state to develop the basic form of a steady-state equation but was cor-
rected to agree with the results of a numerical isodel for unsteady drainage.
Drainage in the cover of a landfill or from the leachate collection system of
an open landfill is clearly transient, and the phreatic surface profile may
differ significantly from the elliptical profile obtained under steady-state
conditions. The profiles will vary while the drainage layer fills and drains.
Consequently, as the profiles vary, the drainage rates vary for -hi sane aver-
age depth of saturation.
The prediction of drainage rates is complicated by many factors. At low
heads, unsaturated flow controlled by capillary action, soil moisture
gradients and gravity can sSynificantly contribute to drainage. Matrix
effects between sell layers can affect the drainage between the layers. Air
can be entrapped in the l«>ers altering the head, hydraulic conductivity and
phreatic surface profile. The unsaturated hydraulic conductivity of soil at a
given moistur* content varies depending on whether the soil is wetting or
drying. Field measurements of hydraulic conductivity, porosity, and field
capacity are difficult to perform precisely and accurately. Soils are
gen-rally not uniform and homogeneous.
This study examines trar.«lent or unsteady drainage from the entire drain-
age layer above a sloped barrier soil layer to verify the equaric- used in rSe
HELP model* This equation was developed by extending equations oevclop»>* in
the literature for simpler cases. This study provided much-needed information
in three areas where the literature is lacking: transient dm. ainage, drainage
from the surface of a barrier soil layer, and drainage from soils above sloped
barrier soil layers. This study measured drainage rates .is a function of
8
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phreatic surface profile for two sands, three slopes, a range of saturated
depths, and two drain spaclr.gs.
DESCRIPTION OF KELP MODEL
The HELP node I Is a computer model chat gererates »ater budgets for a
landfill by performing a daily sequential simulation of water movement into,
through and out of the landfill. The model estimates of depths of saturation
and voluaes of runoff, evapotranspiracion, lateral drainage, and percolation.
Laceral drainage is computed in the model as a function of the average depth
of saturation above the liner, the slope of the surface of the liner, the
length to the drainage collector, and the hydraulic conductivity of the
lateral drainage layer (I):
0.16
2 (0.51 + 0.C0205 a L) K y ' "
u . IU	-|
5 "3L]
L2
(1)
where
Q - average lateral drainage rate for the time period,
inches/day (in./day)
a - slope, diaensiooless
L - drainage length, inches
K • hydraulic conductivity, in./day
>• - average depth of S3turacicn, inches
As presented in the HELP documentation report (I), this equation was
u«.veloped from the Boussinesq equation for unsteady, unconfined laminar flow
through porous media. The Boussinesq equation is obtained by combining
Darcy's law with the Dupuit-Forchheiaer assumptions (also known as Dupuit's
law and Dupuit's approximation) with the continuity equation. Dupuit's law is
a steady-state equation for lateral flow:
q - -X y (dh/dx)	(2)
where
q « flow rate per unit width
y ¦ depth of saturation above the impervious bed
h • height of free surface above the drain
x • horizontal distance from the drain
9
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This law assuae9 that the flow throughout the dep>.h of saturated soil is hori-
zontal, Thus, the equipotent ial lines are vertical and the streaalinas are
horizontal. The second assumption of this law is that the hydraulic gradient
is nqual to the slop# of the :ree surface and does not vary with depth.
The continuity equation for lateral drainage from porous aedia say be
written as fallows
f Csy/31> - -(jq.'ix) + R	(2)
wher*:
f • trainable porosity
C ¦ Ciae
R ¦ rate of vertical infiltration or evaporation
into or out of the saturated soil
Combining Equations 2 and 3. Che Boussinesq equation is obtained:
f (37/aO - 3(K y (3h/ax))/3x + R	(4)
This equation applies Dupuit's law for unsteady conditions.
Equations 2, 3> and 4 were developed for systeas with horizontal or
mildly sloping impervious beds. Landfills typically have liners that are
sloped from 2 to 30 percent, whi*h violates the funs of Equations 2, 3 and 4.
If the flow is assumed to be parallel to the constantly sloped, impervious
bed, then the equipotential lines would be perpendicular to the bed. Under
these assumptions Equations 2, 3 and 4 can bi modified as follows
q	-	-K y cos g (dh/dl)	(5)
f (jy/jt)	- -{sq/al) + R cos e	(&)
f(jy/3t)	- a(K y cos e (3h/ai))/ai + R cos	a	(7)
where
0	" slope of the impervious bed
h • y + I sin b
1	• distance froo drain along the bed slope
Equation 7 is the form of the Boussinesq equation used to develop the HELP
lateral drainage equation and the equation ®olvtd numerically in this study.
10
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To develop the HFL? equation, steady state was assumed	(5y/3t - 0). This
lap Lies that the phreatic surface profile does not differ significantly from
the profile during unsteady conditions, particularly during	periods when the
ohreatic surface is rallir.g, Under steady-state conditions	the infiltration
rate (X) equals the overall average one-diaens ional lateral	drainage rate 'Q; ;
therefore. Equation 7 becomes
Q - K d[y (dh/dl)*/dl	(jj'i
This equation is nonlinear since both y and h are functions of 1. The bound-
ary conditions for this equation are
'.i • 0 at 1 ¦ 0	(9a)
and
dh/dl - 0 at 1 - L	(9b)
where L ¦ length of bed from drain to crest.
Equation 8 vas linearized by setting dh/dl equal to the total change in
head over the total length divided by the total length:
dh/dl - (y + L sin S)/L	',10)
o
where y^ » depth of saturation at 1 - L.
Equation 8 becosres
Q - K (yo + L sin 3)(dy/dl)/L	(11)
Siaildrly, dy/dl was set to equal the average depth of saturation divided by
half of the drainage length:
dy/dl - y/CL/2)	(12)
Therefore, Equation 11 becoaes
C. - 2 K v (v + I sin i)/J	(13)
o
Equation 13 was then fs.-iver.eJ to replace y with a function of y since
yo is unknown in the HELr ecdel. It was also corrected to agree with numeri-
cal solutions of Equation 8 foe perioos when the phreacic surface is falling
after the profile had reached steady state. The correction was xade for satu-
rated depths racing froa 0 to 30 in,, for slopes ranging free : to 10 per-
cent, and for drainage lengths ranging from 25 to 200 ft. The result of this
correction yielded Equation 1 after replacing sir. 9 with a. The slope, a, in
H
-------
diner.sionless form is equivalent to tar. 8, which for snail slopes is
approximately equal to sin 9.
The spatially averaged depth of saturation is not constant in a landfill
with respect to tine, and depending en the rainfall intensity, collection svs-
teo design, and permeability of the soil lavers, can vary greatly in several
hours. Since the drainage rate is a function of the average depth cf satura-
tion, the drainage rate can also vary- greatly in several hours.
As lateral drainage occurs from a drain layer without infiltration, the
average depth of saturation continuously decreases; the drainage rate does
likewise. Therefore, the HELP model solves Equation 1 as a function of time
by applying it for a time step, yielding the average drainage rate for the
tine seep. The tine steps were 6 hr for lateral drainage from above the top
barrier soil layer of the landfill and 24 hr for the lower two barrier soil
layers.
The model reports the drainage rate from above each barrier coil layer
daily, To obtain the daily value for the tcp barrier soil layer, the model
averages the computed values from the four time steps. The units for drainage
are inches/day (volume/day/surface area).
The average drainage rate is a function of the average depth of satura-
tion, which is a function of the average drainage rate. Therefore, the model
solves for drainage iteratlvely by assuming the drainage rate, solving for the
average depth of saturation and then solving for the average drainage rate.
If the calculated drainage rate differs significantly^from the assumed value,
a new estimate of the drainage rate is produced and the process is repeated
until the estiva red and conputed values agree within 0.2 percent.
The average depth of saturation is computed by dividing the drair.able
water in the lowermost unsaturated segment in a subprofile by the drair.able
porosity of that segment, and then adding this value to the sua of the thick-
nesses of all saturated segments between that segment ar.d the barrier soil
layer. In actuality, the average depth of saturation in a landfill, ph/sieal
model or a two- or three-dimensional model would be determined by inf-sreting
the depth of saturation over the area or along a path to the drain and divid-
ing by the area or the length of the path. This is the method that was used
in analyzing data from the physical models in ar. attempt to verify the drain-
age equation.
While the model neglects the lateral variation in saturated depth, the
drainage equation used in the HELP model ins corrected to approximate the
numerical solution of the one-diaensicnal Boussinesq equation as draining
occurred. The numerically generated profiles were used to compute the average
depths of saturation which were subsequently used to develop Equation i.
Therefore, use of Equation 1 by the HELP model implies that the profiles are
the same as those generated by the numerical solutions of the Boussinesq
equation.
12
-------
PURPOSE AND SCOPE
The ob*ecclve of this study was to verify the lateral drainage component
of the HELP model and other USEPA regulatory and technical guidance for
leachate collection systems by determining the lateral drainage rate as a
function of the hydraulic conductivity, slope, length and depth of saturation
of the lateral drainage layer in large-scale physical sodels. the measured
average depths of saturation, drainage rates and drainage times in the
physical models were compared with the results predicted by the HELP equations
for vertical and lateral drainage.
Two large-scale physical models of landfill liner/drain systems were con-
structed and filled with a 3-ft sand drain layer overlying a I-ft clay liner.
A 2-in. layer of gravel was placed under the liner to collect seepage from the
clay. The models were instrumented to measure the water table profile, sub-
surface lateral c'rainage rate, water application, runoff and percolation
through the liner. The models have adjustable slope and two drainage lengths,
25.5 ft and 52.5 ft.
Drainage tests were conducted on both models at three slopes—2, 5 and
10 percent. Drainage from two dlfferrnt sands was studied in each model at
each siope. Several drainage tests ware run on each configuration of the
models by applying water as rainfall to the surface of the sand layer and then
measuring the water table along the length of the models and the drainage rate
as a function of time. The water was applied at several intensities and for
several durations.
13
-------
SECTION 3
MODEL DESIGN, PREPARATION, AND INSTR'JMENT.YTION
MODEL DESIGN" AND CONSTRUCTION
Two physical models were designed and constructed to perform the labora-
tory verification tests. The models, shown in Figures 1 and 2, were con-
structed at two different lengths to permit the examination of the effects of
length on drainage. One model was built to have a usable uepth of 5.0 ft, an
inside width of 5.3 ft, and a drainage length of 25,4 ft. The other model had
the same depth and width but had a drainage length of 52.4 ft.
loth models were constructed of identical materials. The base of the
models was constructed of steel soil test cars used fnr mobility studies.
These cars are reinforced 1/4-in. steel tanks that are 21 ft long, 5.3 ft wide
and 2.5 ft deep. The sides of the cars were extended upward 4 ft with marine
grade, 1/2-in. plywood supported by 2-in. angle iron. Silicone sealant was
used to fill the joints and cracks and to seal and prevent leaks.
The models were placed at h 2-percent slope and fitted with supports for
attaching 5-ft screw jacks at the upper end. A jacking structure was built to
attach to either rodel which could raise the end of a fully loaded rrodel. The
jack could increase the slope of the long model to 11 percent and the short
model to 20 percent. The slopes of the models during testing were determined
by surveying.
Three troughs, approximately 12 in. deep and extending across the width
of the models, were placed inside the lower end of each model as shown in
Figure 3. The bottom trough was used to collect seepage through the clay
liner. The middle trough collected subsurface lateral drainage at the bottou
of the sand drainage layer. The top trough collected runoff from the surface
of the sand. The bottom two troughs were constructed of 1/4-in. steel plates
welded to the steel soil test car. The top trough was fabricated of
galvanized sheet metal and bolt•d in place to the plywood walls. The sheet
metal was scaled to the walls with silicone sealant.
The runoff trough was drained to a 5-ft-deep, 32-in,-diameter (dia) sump
tank via 1.5-dia polyvinyl chloride CPVC) pipe. The water level in the tank
was measured with a steel measuring tape before and after a simulated rain
event to determine the runoff volume. The drainage trough was also drained to
a 5-ft-deep, 32-in.-dia susp tank via 1.5-in.~dia PVC pipe. The drainage sump
tanks were automatically pumped out when filled to a height of about 44 in.,
requiring only about 5 minutes (min) to lower the water depth to about 10 in.
The minimum time required to fill the tank was about 4 hr. The seepage trough
14
-------
RllAMf/t OHAtN Kill
« Htll( AtHtN
WAfl* IfVt
HtCOfilifK
*1	Alt>*H
i OHAl*
I AUK
tirvt*i, 0tA.il I
0«4'.V4u(
•*y 7*"v4<
£/*lu*t tMAHwuct*
HtAOIHoS
MC.UtAtO*
"4 C.i'1 AfOO
• iMrrMOl
VAI VI
Figure
L. Plan vlev of experimental components of
the physical
models.
-------
ft AIM CAUT OftlVe TRAIN
structure
RAIN CART
NOZZLE
PIIIOHITIN

SAND
CLAY
FIITCN FABRIC

XiMf(jm«4kl! MfcMJllAMe
FILL MATERIAL XXio*
* wr r™A •" 'vvv¦'¦	^ ^ ^^-".'lCt>0
(T
tippiho »UC*«t
jkMD OB*"'
FiR"re
Cutaway eiAe
» „jral roodt^•
lew of P^0
-------
Rain Cart
Wit*r
Sub d ty
Ho« •
3p'*y Noiz1•
Top of Stnd Lay*
#.3'
Runoll Tfou;h
SAND
S t • • I Cart
V
Top of City L*y«r
GRAVEL
Percolation
Trough
Or sin Pic. t
Figure 3. Cutaway end view of physical model at the troughs.
17
-------
was periodically drained into a 4-15ter (i) graduated cylinder to measure the
seepage volume. The seepage was drained from the trough via an attached
2-in.-dia rubber hose chat was sealed hecw.'sr, measurements.
Both models were fitted with a device t~ spray water uniformly across the
length of the surface of the sand drain layer. This device consisted of a
l.5-in,-dia ?VC pipe aounted on a 12-ft-long aluminum cart that tracked back
and forth on rails that tan the length o- the model. A nozzle that sprayed a
uniform line of water across the width of the model was attached on each end
cf the pipe. Each nozzle sprayed on one-half of the node! as the cart coved
its maxir.ua distance is one direction. The height of the nozzles and the
travel limits of the cart were adjustable to ensure that the entire surface of
the sar.d wa3 sprayed. The water application rate or rainfall intensity was
adjusted by changing the size cf thi nozzles or changing the water pressure
applied on the nozzles. The intensities of flow rates could be varied between
0.02 and 6 in./hr or 0.06 and 10 galJons/ainute (gpa). The intensity during a
test was determined by meaeuring the volume of water applied as a function of
tiae. A stopwatch was used to measure the time ar.d a Badger Meter, Inc.,
Model Reeordall 12 water aeter measured the water volume. The nozzles usee in
thw tests were Floodjet 1/8 K.25 316SS through 1/4 K1B 316SS that were manu-
factured by Spraying Systems Co.
Schematics of the models are shown in Figures 1, 2 and 3. Figure 1 shows
a plan view of the experimental setup, including the layout of the two models,
water and drain lines, instrumentation, and sump tan>s. Figure 2 shows
a cutaway side view of one model including the troughs, drain lines, sumps,
jacking structure and rain cart. Figure 3 shows an cutaway end view of the
model illustrating the placement of the trough and the shape of the model,
TEST PREPARATION
After construction was completed, the models wr>re filled with three func-
tional layers and a layer of fill dirt as shown in Figure 2. The top layer
was a 3-ft sand layer for lateral drainage. The second layer was a 1—ft com-
pacted clay liner designed tc minimize percolation. Ths third layer was a
2-in. layer of pea gravel to transmit the seepage from the clay liner to a
drain. The bottom layer was a lC-in. layer of fill dirt to build up the pea
gravel layer to promote drai-^ge.
Fill material 10 in. deep was compacted in the bottom of the steel test
car to fill the trapezoidal section of the model and to reduce the volume cf
pea gravel required. The fill material was covered with a 30-ail, butyl
rubber membrane, T-16 manufactured by Firestone. The meubrane was glued to
the walls of the steel car to prevent percolation into the fill material. The
glue was 0~580, a synthetic rubber resin dispersed in solvent and manufactured
by Pittsburgh Paint & Glass.
A 2-in. layer of washed creek pea gravel was placed on the impermeable
membrane and in the seepage collection trough. This layer was designed tc
have a high permeability and low storage potential in order to transmit the
snail volume of seepage rapidly to the collection trough. The layer was
18
-------
covered with Bidia Type C3«. filter fabric manufactured by Monsanto to prevent
migration of fines from the clay liner into the pea gravel, thereby protecting
the ability of the pea gravel to transmit seepage, 7h
-------
1 1 5
110 -
3 10S
u
*•%,
ia
n
100

cr
Q
90 t
35
1 3.0
LEGEND
6 COMPACTION EFFORT, CE 12
O COMPACTION EFFORT, CE 21
O COMPACTION EFFORT , CE 58
9 AS COMPACTED IN MODELS
15.5 ta.O 20.5 23.0 25.5 28.0
WATER CONSENT (% dry wt)
30.5
Figure 4. Compaction curves for buckshot clay soil.
20
-------
Two different sands were used ia Che tests, One sand was Reid-Bedford
sand with Windham clay nonuniferxly dispersed throughout. The clay content
was about 10 percent ir the short model and about 13 percent In the long
xodel. The dry density, as placed, was 106 pcf. This is a relative density
of 96 percent, indicating very good compaction. The specific gravity was 2.7,
and the coefficient of permeability measured in the sails laboratory was about
3.5 x 10" cn/sec. The porosity was C. 3?, the aaxirr.ua drainable porosity was
between 0.21 to 0.25, and the wilting point was about 0.07. The grain-size
distribution, of the s ir.d with the clay is shown in Figure 5.
The ether sane was a local, ungraded washed creek sand. The dry density,
as placed, was 107 pcf in the long model and HO pcf in the short model, indi-
cating both were hi-hly compacted. The specific gravity of the sand was 2.7,
and the coefficient cf permeability measured in the soils laboratory was
2.2 x 10 cm/sec. The porosit; wss 0.35 In the short model and 0.36 in the
long sodsl. The maximum drainaole porosity was between 0.24 to 0,28 and the
uilting poiv.t was about 0.C4 to 0.05. The grain-size distribution is shewn in
t igure 5.
INSTRUMENTATION
Description and Installati*n
The lateral drainage rate from the sand layer ves determined by two
devices, one for low flow rates and the other for high flow rates. Low flrw
rater (below 0,08 gpta or C . 05 in./hr for the short r.odel and 0.02 in./hr for
the long acdel) were determined by measuring the drainage volume as a function
of time using a Weathertronics Model 60'. 0 tipping bucket rain gage. The tip-
ping bucket collected thu water from th« end of the ?VC pipa draining t'.e
drainage trough and discnarged the water into the drainage suap tank. A
bypass was placed in th« PVC pipe to divert che drainage directly intc the
suop tank during periods of very high flow races. Higr.er flow ratei {greater
than 0.08 gpm) were determine.! fcv measuring the water level In the drairage
suop tank as a function of time using a Vea:her Measure Corporation
Model F553-A waci.r level recorder powered at 12 volts by a ZLCO Mo-.el 1060S
low ripple batte'-y eliminator and charger.
Pcie-pres'iure transducers were placed in the clay liner vim the trans-
ducer exposed upward toward the sand layer. The face of the transducer was
set slightly b'.low :he euirounding cop surface of che clay, T/j install a
transducer, a hole about 5 to 6 in, deep and 6 in, in diameter was dug in the
clay. The transducer and wire lead were aced in the hole jr.'; the removed
clay was tamped back in place. The transducer was about 2.'^ in. long and
1 in. in dia-neter. The wire lead frcr* Che transducer vas '.la:ed just below
che surface of the cla/ to prevent interference with the 1attral drainage £roo
the sand Ia%er. The trandccer installation is shown in Figure 6.
Six CTtoaducers were placed in the long model ar.d five ware placed in the
short xodel. Consolidation Electrodynamics Corporation Tyre 4-312 pressure
pickups were fitted in a brass body with a porous stone sc the pressure trans-
ducer couU only measure pore pressure, which in thio case was the head of
21
-------
U.I. SltNOAftO SUVC 'irrnlNO IN INCmCI
u.e. iiano.md iicvc nuntcpt
70
HTOR'jnCIf R
BIO 19 70 JO 40 tO 70 100 140*00
50 w
70 E
l 30
(00 too
60 10 i
i
GRAIN SIZE
OS 0.1
N MILLIMETERS
d.os o.ot o.oos o.oni
COBBLES
GRfiVEL
SAND
SILl OR CLflT |
coani. I rmr
csni>sr 1 ncoiun 1 nxc

LL | PL IPI 106
NRI H.X

I.P.CEUP
A Keid-Hcdford S.ind Willi 102 Vlndluni Clay
O Ungraded Washed Creek Snnd
CLRS3IFICRII0N
O SAND (SP), 11K1H/N: WITH GRAVEL
6SII.T* SAND (Sl'-SH), BROWN
ORflDHTI ON CURVE
Figure 5. Grain-size dlfltrlhutlonn of the sand drainage media.
-------
Transducer Lead
1/2* PVC Pipe
Piezometer
SAND
4* Lon v
j HoIIqki
/ P o f o j S
)i- S t o r. e
Transducer
Figur# 6. Sktcch of pliz'jtr-star and trmsdurrr InstaJ lacion.
23
-------
water on the transducer. The locations of the transducers wre app.-oxinately
3, 6, 11, 16 and 23 ft from the lip of the drainage trough in che short model
and 3, 6, 11 , 23, 35 ar.d 50 ft from the trough in the long model, as shown in
Figure 2. The transducers wr.re placed cn Che centerline dividing the width of
the models. The elevations and locations of the transducers were determined
by surveying.
Piezometers were installed in the sand layer to provide a backup method
for measuring the depth of saturation above the clay liner. These measure-
ments were also used to calibrate the pressure transducers and tc check the
accuracy of the transducers throughout the testing. Piezometers were con-
structed by Attaching 1.5 in.-dii, 4-in. long-cylindrical porous stones to
I/2 — ir> ,-dia ?VC pipes of approxinately 4-ft lengths. The piezometers were
installed with the; stone end placed flush with the tcp of the clay liner and
with the pipe extending straight up through the sand layer. The depth of sat-
uration was determined by treasuring the depth to the water surface from the
end of the pipe and then subtracting that v*i„e from the total le

-------
bucket. The number of tips was counted and compared with the number r-scorded
on the water level recorder. The drainage rates computed froir the tipping
bucket data and the water level recorder data were compared at the oid/ang* of
the flow rates and agreed well with each other.
Seepage was collected in the xcdel for a week or longer anJ then drained
into a 5-gal bucket or a graduated cylinder. The volume cf seepage was mea-
sured with a graduated cylinder.
The depth of saturation above the clay liner was treasured in two ways—
manually using piezometers and automatically using pore pressure transducers.
The piezometers were calibrated by aeasuring the ler.gch cf each piezometer and
surveying the location and elevation of each piezometer with respect to the
lips of the runoff and drainage troughs ana the clay liner. Tl.e end of each
piezometer vaj placed flush with the surface of the clay liner.
Prior to installation, the transducers were calibrated in the instrumen-
tation laboratory to determine the linear response factors for a unit increase
in head and the readings fir zero head. The responses from the transducers
were recorded using a Digi'.rend Med el 210 digital recorder with a Hewlett
Packard Model 6113A 5-volt EC power suopiy, a Doric Model 214 digital clock
and a Doric Model DS-100 integrating microvoltceter. A Consolidated Electro-
dynamics Corporation Type 8-108 DC bridge balance was used to convert the
millivolt signals from the transducers to pressure readings in pounds per
square inch for the experiments with the Reid-Bedford sand. This same equip-
ment was used both in the instrumentation laboratory and in the rodel tests.
After Installation, the locations and elevations of the transducers were
surveyed with respect both to the l^s of the runoff ard drainage troughs ind
the clay liner. After the sand was placed above the transducers, the trans-
ducers were calibrated again in place to establish the readings which corre-
sponded tr> iero head of water on the clay liner since the transducers were not
placed flush with the surface of the liner. The transducers were also checked
to determine if the linear response factors for a unit increase in head were
the same rs in the instrumentation laboratory. The zeros and response factors
were determined in place by saturating the sand layer and flooding the surface
to establish free water suffices of known elevations r.bove the sand. Using
the surveying data and the linear response factors, the zeros were calculated.
The linear response factor.: were checked by computing the changes in the
readings for several known changes in water level and comparing them with the
factors determined in the instrumentation laboratory. Vhe response factors
ag.eed well with the instrumentation laboratory data but the zeros varied
considerably.
The piezometers were read periodically throughout the testing to provide
a backup for the transducers. The readings of the piezometers were compared
with those of the transducers to better define the zeros for the transducers.
After testing was completed, corrections to the zeros were determined from the
comparisons. The corrections ranged from about -2 to 2 ir.. (±5 percent of
maximum), though half of the transducer readings required little or no correc-
tion. The zero of one transducer drifted with time in a constant manner, and
the correction was made js a function oi time. After correcting the readings
25

-------
of che transducers which had piezometers associated with then, the readings of
the other transducers were corrected. These readings were corrected by
plotting the readings to form a profile of the saturation at about fi\e dif-
ferent average heads for each test, A smooth curve was drawn through the pre-
viously corrected readings and zero at the trough. The offset of the other
readings for each pijt was noted to dcterair.e whether the offset was constant
at all average heads and for each test. A constant offset at all heads ar.d
for each test indicated that the zero was off ar.d required correction equal to
the offset. If che offsets changed continuously, particularly in one direc-
tion, it would indicate that the readings drifted. Readings of only one
transducer drifted and the transducer was located at a piezometer. If the
offsets increased or decreased ur.iionnly with increasing average head, it
would indicate that the linear response factor was incorrect. Hone of the
linear response factors required correction. Nonsysteaatic changes in the
offsets indicated the variance in the transducer readings. The standard devi-
ations of che readings were abouc *0.25 in.
26

-------
sect:jk -
EXPERIMENTAL DESIGN AXT? PROCEDURES
EXPERIMENTAL DESIGN
A coirpiete block experimental design was used tc examine the efiects of
drainage length, slope, hydraulic conductivity, depth of saturation, rainfall
intensity and rainfall duration on the lateral subsurface drainage rates. The
block design was selected because it provide;," the nost data with the least time
and expense for construe tier, and ncdel preparation. Only two nodels cculd be
built due to their costs, and only tve sands could be exacinec due fo the
unavailability of another of significantly different permeability. Several
slopes and rainfall events could be examined quickly since very little tine
was required for changing these test conditions. Also, the time requirements
and costs for running ar, additional test with a different slope or rainfall
were less than LQ percent of the requirements for preparing the r»odel for a
different san-j, Additional rainfall events were examined in lieu of
replicates since the lateral drainage rate ss computed by Equation 1 does not
directly consider the effects cf rainfall intensity cr duration. Also, sir.ee
a complete block design was used, the effect of a change in a variable is
directly exaoir.ed under multiple test conditions, reducing the need for
replicates.
A sunr.ary of the rcdel tests is presented in Tables I, 2 and 3. Two
drainage lengths were compared—25.4 ft and 52.i ft. Three slopes were
examined—approximately 2, 5 a^d 10 percent. Sards of two hydraulic con-
ductivities were used—4 x 10 cc/sec (fine sand) and 2.2 x 10 c./sec
Ccoarse sand). Four rainfall events were examined—a 1-hr rainfall at
0,50 in./hr, a 2-hr rainfill at 1.50 in. ,'hr, a 6-hr rainfall at 0.50 in./hr
and a 24-hr rainfall at 0.125 in./hr. Also, water vae applied to the sand for
a long period of time (generally more than 36 hr) sc a rainfall intensity
that would maintain the average depth of saturation in the sar.d at 12 in. In
addition to these drainage tests, the sand vas sacurs'ed, predominantly from
the bot tea up for several test conditions, and then allowed to drain.
The order of the testing was arranged to accocr.odate xanpover and equip-
ment restrictions. Ail testL were performed on a given sand before replacing
cne sand since it would have been very costly and tice-consuming to continually
change the sand. Also, the placement of the sand would not have beei' identical
each time, which would have been a source of error in the comparisons of the
other variables such as drainage length and slope.
Ail rain events were performed for a given test condition before changing
the slope of the model. This was done to minimize manpower requirements ard
27

-------
TA3LE 1. EXPERIMENTAL CONDITIONS FOR UNSTEADY DRAINAGE TESTS WITH RAINTALL



Model

Painfull
Rainfall
Rainfall
Test
Type of
Length
Slcpe
Intensity
Duration
Volume
Setut)
Sand
(f c)
(") '
(In./hr)
(hr)
(gal)
1A
Fine*
25.45
1.8
o.sa
1.0
48.5
1»
ft
It
II
1.69
2.0
284,2
IC
It
ft
It
0.56
6.0
284.4
ID
If
f*
It
0.14
24.1
287,0
IE
ft
It
It
1.69
2.0
282.2
2A
11
II
4.9
0.58
6.0
291.6
3A
It
II
10.4
0.56
1.0
48.6
3B
i»
If
It
1.74
2.0
291.7
3C
it
It
If
0.57
6.0
291.1
2D
it
11
M
0.14
24.0
291.0
4A
it
52.45
2.1
0.57
1.0
98.5
4B
it
It
It
1.70
2.0
588.0
4C
it
It
tf
0.54
6.3
593.0
',D
ft
tl
11
0,14
23.5
587.0
Ik
ft
It
5.0
0.57
6.0
591.0
6A
f»
• t
10.3
O.S7
1.0
98.6
6B
ft
ft
H
1.70
2.0
589.2
6C
ft
*1
II
0.54
6,2
581.0
6D
ft
ft
II
0.14
24.6
592.2
?A
Coarse**
25.45
1.9
0.58
1.0
48.6
7B

It
II
1.73
2.0
291.6
7C
»»
If
II
C, 56
6.0
291.6
7l>
ft
It
It
0.14
27.9
338.7
71
It
It
II
0.14
24.2
290.9
8a
1*
It
5.4
1.73
2.0
291.4
BB
it
It
It
0.59
5.9
291.1
9a
II
It
10.0
0.58
1.0
48.6
9s
t»
1*
ft
1.73
2.0
291.0
9C
It
It
It
0.58
6.0
291.6
SD
It
If
11
0.12
29.6
290.9
9E
ft
It
If
0.14
24,0
291.0



(Contin'
je^)


28

-------
TABLE 1, (CONCLUDED)


Model

Rainfall
Rainfall
Rainfall
Test
Type of
Length
SI -)oe
Intensity
Duration
"o;uce
So cup
Sand
(ft)
(%>
(in./hr)
(hr)
(sal*
I OA
Coarse**
52.45
2,3
0.57
1.0

1GB
11
629.4
II
1.70
2.0
539.2
IOC
(I

II
0.56
6.1
591.0
10D
ft
It
II
0.14
24.0
591.0
11A
tl
It
5.2
1.62
2.1
589.4
i IB
II
f)
tl
0.57
6.0
590.4
12A
*1
II
9.9
0.57
1.0
98,6
12B
If
H
H
1.71
2.0
589.2
12C
tl
II
tl
0.57
6.2
581.0
12D
ft
I*
It
0.14
24.0
590.4
12E
~1
It
1!
1.70
2.0
589.4
* Hydraulic conductivity ¦ 3
x 10 ^ cm
/sec


** Hydraulic conduct
ivity - 2
.2 x JO"1
cm/sec



TABLE 2. EXPERIMENTAL
CONDITIONS
FOR STEABY-S
TATE DRAINoGE
TESTS





Avg.




Model

Depth of
Rainfall
Drainage
Test
Type of
Length
Slope
Saturation
Rate
Rate
Setup
Sand
(ft)
(")
(in.)
(in./hr)
(in./hr)
1
fine*
25.45
1.8
12.6
0.0446
0,0336
2
if
II
10.4
13.2
0.0714
0.0543
3
fl
52.45
10.3
12.7
0.0589
0.0249
4
Coarse**
25.45
1.9
U .4
0.0928
0.0708
5
It
»»
10.0
11.3
0.2070
0.1688
6
If
52.45
2.3
11.3
0.0464
0.0230
7
It
M
9.9
11.8
0.1212
0.1050
_3
* Hydraulic conduct:'-"ley - 3 x 10 em/sec
** Hydraulic conductivity ¦ 2.2 x 10 cm/sec
29

-------
TA3LE 3. EXPERIMENTAL CONDITIONS FOR DRAINAGE TESTS USING
PRESATITRATED SAND
Test
Sertrs
r>-p» of
Sand
Model
Length
(in.)
Slope
Initial Avg, Depth
of Saturation
	(in.)	
1
2
3
4
5
6
7
8
9
10
U
F Lne*
Coarse**
305.4
629.4
305.4
629.4
8
I
LQ
20.95
16.79
27.35
24.74
9.24
29.79
19.14
2.44
27.75
21.18
14.62
* Hydraulic conductivity
** Hydraulic conductivity
3 x 10 |tn/sec
2.2 x 10 cm/sec
because It uculd have been difficult to achieve exactly the same slope again.
The order in which the slopes were tested was not constant to accomodate
manpower and equipment scheduling. The scheduling introduced sone randomness
into the order of testing the effects cf slope. Similarly, the order of the
rainfall events for different test conditions was varied to accommodate the
work schedules for personnel. This scheduling introduced seme randomness into
the order of testing the effects of differences in rain events.
TEST PROCEDURE AND DATA COLLECTION
Each test was run in one ot three manners. Lateral drainage was measured
from a presaturated sand; from a sand before, during, and after a simulated
rainfall event; or from a sand while attempting to maintain a constant aver.age
depth of saturation of 12 ia. The test procedures for the three manners vere
very similar; only the procedure for applying the water to the model varied.
The procedures for running a test vete as follows;
30

-------
1.	Prepare model for testing. This included preparing the sand by plac-
ing or leveling; preparing the rainfall cart for desired rainfall intensity by
changing nozzles, nozzle heights and end stops; changing the slope of the
model; draining sump tanks; and preparing the recorders.
2.	Initialize test. This included starting the recorders, setting the
time, draining the lateral drainage trough, recording tect conditions, measur-
ing initial depth of water in runoff sum? tank, and recording the initial
water meter reading,
3.	App]y vater to the model as prescribed for the test. This Included
adjusting the water application rate and the spray pattern.
Note peculiarities cf the test including unusual environmental condi-
tions, problems, and variations from the test procedure.
5,	After the rainfall was completed and runoff ceases, measure the final
depth of water in runoff sump tcnk.
6.	Measure water levels in piezometers periodically.
Drain tanks as required.
8.	Continue testing until the heads become so low as to nearly expose a
transducer to air and then stop drainage or start a new test.
9.	Collect data from recorders and reduce the data.
Prior to calibration of the pore pressure transducers, water was added
slowly to both ends of the model. Ir. this manner the rand was saturated pri-
marily from the bottom up, which forced rtie air out oi the sand. Following
the calibration, the sand was drained and the drair.^ge rate and heads were
measured continuously while ^raining. Drainsge from presaturated sand was
measured for only four test ;onditiotis since saturalicr. was required only for
calibration.
For the tests in whi-h a constant average head of 12 in. was aaintained,
water was applied to the model to increase -he head rapidly to 12 in. This
water was applied at a rate of about 0.5 ir,,/hr. After reaching a head of
12 in., the rainfall intensity wa3 decreased to the intensity estimated as
necessary to maintain the head. The desired intensity was determined by
adding the evaporation rate, the rat# of leakage and seepage, and the drainage
rate which corresponded to an average head of 12 in. during a previous rain-
fall for the same physical test conditions, '."he intensity wa« adjusted during
the test if the head did not r«*""-i.n constant near 12 in. The rain was con-
tinued for at least 12 hr aftai the hr.ad remained nearly constant.
The evaporation rate ranged from about 0.2 to 0.3 in,/Jay while raining,
as estimated by the differences in the water application rate and the lateral
drainage rate during the steady-state drainage tests. Th« rate was dependent
or. building temperature and on the spray nozzle. The building temperature
31

-------
ranged from aboot 58*F co 9Q*F, but for most tests the temperature ranged
between 70°F and 82®F,
Leakage varied directly as a function of head. Both models leaked, but
the short model was more watertight than the long model. The leakage rates
for the short model were measured to be about 0,02 in./day at a head of
20 in. and about 0.01 in./day at a head of 12 in., while the leakage rates for
the long model were measured to fce about 9.12 in./day at a head of 20 in. ar.d
about 0.0? in./day at a head of 1-' ir., Neither model leaked at heads below
6 in. since all of the seams in the model were above this depth. The long
model leaked more because It suffered greater deflections upon loading due to
its greater length. This caused more seams to rupture and leak. These
leakage rates we-e used to correct the water budgets for each test.
The majority of the drainage tests encompassed measurements before, dur-
ing, and after a specified rainfall event. In these tests, a specified quan-
tity of water, either 48.5 or 291 gal for the short model and either 98.5 or
591 gal for the long model, was sprayed across the entire surface of the sand
in a specified period of time—1, 2, 6 or 24 hr. These quantities of water
correspond to about 0,57 in. and 3.4 In. of rain, respectively, for both
models. During these tests, the head increased from a starting point of about .
4 in. to a peak value occurring about 30 sin after the rainfall ceased, and
then returned to about 4 in. when the test was stopped. Several tests were
allowed to continue until the head was nearly zero inches.
DATA REDUCTION
The readings of the pressure transducers were recorded on a digital
recorder at an interval ranging from 10 to 50 tain, The reading for each
transducer was converted into head of water at the location of each trans-
ducer. The reading was converted by subtracting the zer<-> for the transducer
and then multiplying this value by the linear response factor for the trans-
ducer to convert the millivolt value to pressure head in inches of water. The
head was then corrected for changes in the barometric pressure and temperature
using the readings from a transducer installed to monitor these changes. The
heads were then corrected for the differences between the elevations of the
transducers and the elevations to the surrounding clay liner. After the
actual heads were computed, the heads throughout each model at each time
period were averaged using the trapezoidal rule. The head at the lip of the
drainage trough was assumed to be zero, and at the upper end the head was
assumed to be the same as at the uppermost transducer.
The volume of drainage was r- corded continuously on a strip chart. The
drainage rate was determined by reading the change in volume of drainage
collected in the sump tat* for an interval of time or the volume of drainage,
as tip3, passing throw.h the tipping bucket during an interval of time. These
volumes were then divided by the interval of time to yield the average
drainage rate. This rate was recorded as the drainage rate at the center of
the time interval. The intervals ranged from 15 rain to 3 hr depending on how
rapidly the drainage rate was changing, l.ie drainage rate was then correlated
with the heads by matching the time of the readings.
32

-------
SECTION 5
PHYSICAL MODEL RESULTS
This section presents the results of the drainage tests. These results
include measurements of the saturated depth as a function, cf distance from
drain, measurements of saturated depth and drainage rate as a function of
time and, similarly, measurements of drainage rate as a function of saturated
depth. Explanations of these results are presented in the next section alofg
with comparisons between these results and the HELP model predictions.
SATURATED-DEPTH PROFILES
The depth of saturation was measured using pressure transducers at various
distances from the drain. Plots of depth of saturation versus distance f rotn
the drain are shown in Figures 7-10 for various times during the experiments.
The depth measurements at a given time were used to compute the average
saturated depth (y) for that tiae by the trapezoidal rule. The values of y
are noted in the figures for all saturated-depth profiles shown, a comparison
of profile shapes indicates significant differences between the rising
saturated-depth profile (fill conditions) and the falling saturated-depth
profile (drain conditions) for the same y. The profiles are steeper near the
drain when filling than whea draining. Th«? difference is greater fcr higher
infiltration rates. Profile plots are shown in Figure 1.4 for steady-state
discharge at slopes of 2 and 10 percent (a. • 0.02 and 0.10) when y equaled
approximately 12 in. Steady-state profiles are very similar to the profiles
for draining. Similar plots for the hydraulic head above the drain profiles
are shove in Figures 12-16. All profiles shown in Figures 7-16 weris t^ken
from the 6-hr rainfall experiments, except for the fine sand profiles in the
short model at 2-percent alope, which were taken from the 24-hour rainfall
experiment.
DRAINAGE'RATE AND SATURATED DEPTH
The variations in average saturated depth and drainage rate with time art
shown in Figures 17-22. Again, these figures represent 6-hr rainfall experi-
ments with the exception noted above. In some cases, the entire thickness of
the drainage layer approached saturation prior to the end of rainfall. As a
result, the curve of average depth versus time for a few cases approaches a
plateau near the point of maximum average 
-------
*9
35
3a
«	»y-14 sa* riLL
a—Oy-2S.84* DRAIN
9-14.78 * DRAIN
~—•5- 9.86* DRAIN
2* SLOPE
"O
25
20
IS
ia
5	10	IS	2P	2%
DISTANCE TROW DRAIN * (FEET)
#—*y»<2.32* fill
a—05.21,88' drain
~—« crem
Figure ?. Saturated depth profiles for unsteady drainage during 24-hr
rainfall teat at 2-percent slope and during 6-hr rainfall teat
at 10-percent slope using fine sand in short model.
34
-------
s
i
T
y
R
A
t
£
D
D
£
P
T
H
2* SLOPE
a—Ov-2S.9r Of:AIN
~ y-tS.13" DRAIN
•—*¥- 7 23* yRAIN
23 -I
zs 3a
DISTANCE FRCf DRAIN
trerrj
10* SLOPE
y-fS.*»*
FILL
DRAIN
DRAIN
DRAIN
2! .?! *
IS.S2
a.M*
a ~
19
I
IS
T
25
29 25 30
DISTANCE f*R0N DRAIN
I
35
I
*0
crem
—r~
45
1
sa
S.'i
Figure 8.
Saturated depth profiles for unsteady drainage during 6-hr
rainfall test using fine sand in lcng oodel at slopes tf
2 and 10 percent.
35
-------
30
2* SLOPE
fill
DRAIN
CRAZH
2,33* DRAIN
zs
20
IS
10
s

a
15
6
19
a
DIS~ANC£ FTON WU1N * iFECf)
s
A
t
U
R
A
T
£
D
0
£
P
T
M
I
N
c
M
r
2fl
IS
10 -I
y- 4,it"	FILL
-"v-ta.ao*	DRAIN
" V- ~ a?*	DRAIN
-*y~ a.SB*	DRAIN
10% SLOPE
Figure 9.
DISTANCE r*C« DAAIM » CFtTTi
Saturated depth profiles for un»ttady drainage during
6-hr rainfall east using coarse sand in short aodel
at slopes of 2 ar.d 10 percent.
36
-------
2* SLOPE
*—*>-13. *9'	rju.
a-"0 9.18,S! •	OTAZN
13,34' DRAIN
•—•*- *.88* CRAXN

>S 29 2S
sa
s
3£
46
ss
e
distance fwcm au» * cram
0% SLOPS
tt—a	6. 78* r.LL
3—88* ORAlN
v-	S.S8* DRAIN
•—*V-	7 ?«• 5RAIN
®—1 .98* DRAIN

¦Ok.
3£.
SS
e
15
44
9
DISTANCE IHKCN WAIN * C«ETJ
Saturated depth profiles for unsteady drainage during
6-hr rainfall test using coarse sand *,n long model at
alopea of 2 and 10 percent.
3?
-------
s
A
T
U
"»
*
T
E
D
0
£
P
T
M
I
N
C
M
E
S
29
16
10
SHORT MOOEL
.314. r3¦«. i n.nxe
o-a.82t.COARSE
•»«.ass.coarse
-T_
is
is
i
39
—r~
2S
r-
38
—r-
3S
T
l
¦4S
m
OISTamcC PRlSl MUXN * crtTT>
Figure 1t.
S«irurat«d depth profiles for steady-state drainage
in both physical models.
38
-------
"3S
?S
S
0
3
S
9
OIS*ANC£ FRON DRAIN , :FEE~2
*—nu.
9<1 " SPAIN
»4.3S* DRAIN
•—•N-37.I3* DRAIN
13* SLOPE
DISTANCE FROM DRAIN * CFCm
:ig*jre 12, Head profiles for unsteady drainage during 24-hr rain-
fall test ac 2-percent slope and during 6-hr rainfall test
at 10-percent slope using fine jaid in short model.
39
-------
*—"5-21 .78*	FILL
a—o H-.32,22 *	ORAJN
*"~ZI . 74*	5RAIN
*-13.90*	CRAIN
2% S
2S 39 3S *a
distance fson draim » tfrrrj
*—•h-47.73* FILL
B—0^53.95'	DfUlN
~—»fi-4?,78*	DRAIN
»R-4l .82*	PRAlN
10% 3
IS 28 ZS 3a 3S
DISTANCE nw» DRAIN > CFECTi
Figure 13. Head profilts for unsteady drainage during
test using fine sand in long aodtl at slope
10 p«rc«nt.
40
-------
*	•h-10.67* rlLL
a—e^is.-M* DRaIN
DRAIN
DRAIN
2S
.a-'
29
S
0
S
2* SLOPE
a
i
is
25
29
a
19
DISTANCE FROM DRAIN * CFEET3
?s

39
2S
28
IS
jt'
10* SLOPE
s
9
2S
te
28
S
19
a
distance from drain * crcrrj
cigure 14. Head profiles for unsteady drainage during 6-hr rainfall
teat using coarse sand In short node.', at slopes of 2 and
10 percent.
41
-------
-to
r
35 J
a.8-4"	fill
3—BZ.2C.7S'	DRAZN
~ h-23.se*	ORAIN
•—•R-12.13'	DRAIN
.-a

20 -i
2% SLOPE
n	1	1	1	1	r
10 IS Zli 2S 30 3S 40 *6 S3 SS
DISTANCE FROW DRAIN * CTEST2
N
E
A
D
A
8
n
v
e
D
R
A
I
N
ea
78 i
«—*>y*3fl,97*	rUX
B—OFr-4S.68*	DRAIN
•—**-.<0.68*	DRAIN
*—*h-38.3«*	DRAIN
•—•N-Sa.SS*	DRAIN
10* SLOPE
DISTANCE FROM DRAIN * 
-------
*—*<*»0 ai8,«ThC
a—ca-0.104, fine
~ a-0 . 85 S, COARSE
0 '00.COARSE
.M-
3S
jr'
30
25
S
a
SHORT MOOEL

s
a
5
•S
a
18
DISTANCE FROM DRAIN * CFEET5
ias, fine
a—« o-0 .023. CCARSE
~—* a*9 . as« , COARSE
LONG MODEL
SS
distance from drain » crcrr>
Figure 16, Head profiles for steady-state drainage in both
physical models.
43
-------
80
2i
FINE SAND
AVtRAQC BICTM
8RAINAOI NATS
20
Hi
z
o
s
- 30
z
k.
w
a
HI 20
C
10 <
*
t;
•d
>
«
.03
10
.00
10
20
40
0
TIM I PROW START OF RAIN (HOURS)
HI
z
w
z
~»
ui
a
o
*
as-
20-
1»-
10-
-o.s
COARSE SANO
Figure 17.
• AVIRAM 6t»TN
O ORAINAOf RATI
tL
a
<
s
ui
o
m
a
¦0,1
"5.0
¦ I M i I I < | • I n • • I I • > . ¦ 		J<1
-------
so-j
	'	
FINE SAND
• AVKAA4B BSPfH
0 ORAiMAOl RATI
TIME PKOM STAHT OP RAIN 'HOU*S>
l-o .ae
-4.1 $ Ml
o.as
-0.00
21-
10-
X
u
X
1.
M
a
m
9
<
c
Ml
»
m
COARSE SAND
• tvciuaa oiPfH
B DRAMAS* RATI
-0.#
-0.4
30	30
Tim FROM START OF RAIN (HOURS!
Figure 18. Average saturated depth and drainage rate as a function of
tine for 6-hr rainfall test using both sanda in short
model at 5-p«rcent slope.
45
-------
so-
-4
*«UAU oarm
PINE SAND
SRAIKAat RATI
40
30:
10
20
40
10
0
TIMC MOM START Of KM* (N0U*9>
a*
AVtlAOt IIM*
COARSE 3AN0
10-
OKAIHAdt »4T1
m
m
X
0
1	1«
*1
o
10
kl
0
<
1
kJ
10
0
20
40
90
ae
TIMC MOM STAl'T OP (AID (HOURS)
Figure 19. Average saturated depth and drainage race as a furct:
tine for 6-hr rainfall test using both sands in shor<
aodel at 10-percent slope.
46
-------

SO
AVINAM 9IPTH
BKAIHA&g RATI
40
30
m 30
10
I^IIIIHIIt|IHIIMII|l
10
«0
30
40
?0
BO
80
7mi pnom start or rain fwouiits
s
u
«
X
o
«
c
w
>
mhMm
• 4VKRA4I QtPTN
COARSE SAND

iao 140
TIMS MO* $TMT OP RAIN t»OUX*>
Figure 20. Average saturated depth and drainage rate as a function o
tiae for 6-hr rainfall test using both sands in long node
at 2-percent alop«,
47
-------
.Ill I
FINE SAiO
	'	
• AVIRAOI 6IPTM
Q t)««IMAOI »ATI
Ml 11 I
40
T*»
>0
•0
TO
'^.2
«T
to
TIME MOM JTi'«T Of RA»K (HOWiSS
» AVIMA9I 6«»tw
o »n*iM*ai Miiti
COARSE SANiJ
TIME PROM START OF RAIN (N0UR9)
Figure 21. Average saturated depth and drainage rate as a funetio
clae for 6-hr rainfall test uslrg both sands in long a
at 5-parcent slope.
48
-------
,23
to
.««¦•<«! depth
fine ;and
(Ktmidl S«T1
20
30
0.18 m
0.10
10-
0,00
Baa^pMHPaaaaa,iBHVv
10
0
20
30
40
TIME FROM snm OP RAIN '.HOURS)
28
20
Z
o
z
18-
X
k.
s "»•
«
K
m
T-r-
0
COARSE 3AN0
• *vi«*a« etPTM
a dramas* »ati
10
"~r <¦
20
30
40
Tine FROM START OP RAIN (HO'JRS)
-0.4
¦0.4
Figure 22, Average saturated depth and drainage rate as a function of
time for 6-hr rainfall test using bot* sands in long model
at 10-percent slope.
49
-------
a 175-
SHORT MODEL.
3
Q
£
a ' sa-
« 125-
£	J
e e i ®e-
u
• A7S-
<	*
*	3
Z	a as»-
J.	j
<	¦<
X	4
o	a 925-
1
a aa#-
' a	>5	20	25
AVERAQE DEPTH (Inch®#)
3a
3
O
•>»
o
c
UJ
H
<
OS
<
DC
Q
a ao-
a as-
LONQ MODii
a a?
a ae-
• as-
a a-t-3
a 13
a az-
a ai-
a aa-v.
s	i a	6	?s	aa	as
AVERAGE DEPTH (lnch««)
pigur« 23. Unsteady drainage rate aa a function of average saturated
depth for fine sand in both physical aodels.
50
-------
fchORT MODE
? a i s
» Q(
x •
I
O a 83-
a
a e ia e
AVESACE D£?TH (lnch««)
i i J8
a «zs<
a ase-
LOMG MOOGi.
a	;
5 j
« » '®«i
e
u
B
Z- a »?s-
ui
<
ec
z
<
It a 82S-
Q
3
a a-aa-L

a z « e a ia * ia e ia 22
AVifl,v:iE DE?TH (Inehaa)
Figure 24, Unsteady drainage race- ?.? a function of average saturated
depth for coara-i sand in both physical aodels.
51
-------
coarse sand. Similar hysteresis occurrid ^ith th*s fine sand. The da-.a for
Che filling cycle were not plotted becausa they crossed over the Iir.es for
higher slopi. For the same y, the dnimga rate during filling is greater
than during draining. This is consistent with the profiles in Figures 7-16,
which show steeper hydraulic gradients near t.ie -irain for filling conditions.
The curves also show that drainage continues after y has essentially re?-hed
zero. This is presumed to ae drainage of capil'.ary water, commonly called
delayed yield. As estimate of this capillary water volume when 7 had just
drained to 0 in. based on an analysis of the experimental data is about
0.1 in. (cubic inches per square inch) for the fine sad and C.3 in. for the
coarse sand.
DRAINA2LE POROSITY
The y and drainage r-te data presented in Figures 17-22 were used co cco-
pute drainable porosity at various heights within the tiodel. The volume of
drainage water collected while the average saturated djpth fell from y, to
y,, was divided by the total /olusa contained in the model between y, af.d y,.
TRis cumber was used as an estimate of the drainable porosity within the
region between y, and y?. The results are shown in Table 4. It is apparent
that the drainable porosities in Table 4 (ranging from 0.01 to 0,20) are less
than those cited in Section 3 (ranging from 0.21 to 0.28). Tha latter velues
were naiHtnmi values based on soil moisture and porosity measurements taken
after all experiments were concluded. The values in Table 4, particularly for
high j values, are low due to delayed yield and capillary effects resulting
from the high drainage rate. Nevertheless, significant reductions in drain-
able porosity appeared to exist with increasing height above the clay liner.
A closer examination of these data using regression analysis resulted in the
following predictive equation for drainable porosity (IjP) at a point located a
vertical distance y (in inches) above the clay liner:
DP - DPC (0.955 - 0,0327 y)	(14)
where DPC » drainable porosity constant. Values for DPC are iisxed in Table 5
for eight experimental cases. Based on an examination of the data, a lower
limit for DP was set for this study at 0.02?i DPC. It is assumed that the
presence and vertical distribution of entrapped air were primarily responsible
for the change in drainable porosity with height, although no measurements of
entrapped air were obtained.
Table 6 shows the time for y to fall from one level to another. As
expected, the time for y to fall from 18 to 12 in. was significantly less than
from 12 to 6 in. because of the larger discharge rates at larger y values.
However, the smaller drainable porosities at latger y values also contributed
to the shorter drain times. For the coarse sand experiments! the times for y
to fall from 12 to 2 ia. ranged from 0.5 to 1.4 days for the 25,4-ft-iong
model and from 1.1 to 4.0 days for the 52.4-ft-long modal. The fine sand
experiments experienced much lor ,er drain times and therefore were not
monitored long enough for y to reach 2 in. In all experiments, y drained from
18 to 12 in. in less than about 1 day.
52
-------
TABLE U. DRAINAB1.E POROSITY
Uralnable Porosity Between y. and y„
Type of Slope
Sand	(Z)
Fine
Fine
Fine
Fine
Fine
Fine
Coarse
Coarse
Coarse
Coarse
Coarse
Coarse
2
5
10
2
5
10
2
5
10
2
5
10
Drain
Length
(ft)
y, - 30
y2-25
y, - 25
v„ - 20
y j - 20
y„ - 15
y, - i>
y2 -
25.A
25.A
25. A
52.A
52.A
52.A
25.A
25.A
25.A
52. A
52.A
5?. A
0.029
0.017
0.017
0.007
0.010
0.020
0.015
0.016
0.015
0.020
0.015
0.050
0.039
0.052
O.OA'j
0.0A7
0.058
0.088
0.09A
0.093
0.117
0.099
U.G88
0.09A
0.085
0.1 3
0.090
0.131
0.127
0.13?
y, - io
0.137
0.109
0.151
0.1 J3
0. 109
0.136
0.161
0.153
y, - 5
P2 - 2
0.187
0.186
0.171
0.209
0.201
0.185
* Values for y^ and y^ expressed In Incites.
-------
TABLE 5. VALUES FOR DRAINABLE POROSITY CONSTANT, r~C
Typ« of
Sand
Slope
(2)
Drain
Lftigth
fft)
DFC
Fine
2
25.4
0.16
Fins
10
25.4
0.14;
Fine
2
52.4
0,151
Fine
10
52.4
0.14i
Coarse
2
25.4
0.21!
Cjar^e
10
25.4
0. 191
Coarse
2
52.4
0.22<
Coarse
10
52.4
0.20(
54
-------
TABLE 6, DRAINAGE TIMES*
Tiae co Drain from y, to v (hrs)
i	* i
Type of
Sand
Slope
(2)
Drain
Length
c-:t)
v, • 18**
y2 - 12
y, - 12
y2 - 6
yl " '
y2 - •
Fine
2
25.4
11.0
-
-
Fine
5
25.4
10.5
16.5
-
Fin*
10
25.4
8.0
15.8
-
Fin*
2
52.4
24.4
>50
-
Fin®
5
52.4
21.6
>37
-
Fine
10
52.4
16.2
-
-
Coarse
2
25.4
7.0
12.1
20.6
Coarse
5
25.'.
4.3
9.2
10.9
Coarse
10
25.4
-
5.0
6.2
Coarse
2
52.4
19.0
36.5
60-5
Coarse
5
52.4
8.7
21.5
27.0
Coarse
10
52.4

12.0
i 3.9
* Drainage tinea la this cable were cakes from 6-hr rainfall experiments,
exceet for che fine sand, 2-percenc slope, 25.4-ft drain langth case which
was taken from a 24-hr rainfall experiment. Sashes indicate that data
were cot available^
** Values for y ^ and expressed In inches.
55
-------
SECTION 6
VERIFICATION OF THE HELP MODEL
Comparisons between the HELP model predictions aaJ Che actual measure-
ments are made ia this section to assess the accuracy of the HELP lateral
drainage equation. All parameters required to compute the drainage rate by
the kEL? equation except the hydraulic conductivity were measured for each
drainage test. Hydraulic conductivity measured in a penseaaeter in the soils
laboratory differed significantly from hydraulic conductivity calculated from
data on drainage rates and depth of saturation measured in the large-scale
physical models. This is thought to be due to differences in placement, com-
paction, and preparation of the sand drainage media, and due to entrapment of
air in the sand. As described in the documentation report for the HELP model
(1), the lateral drainage equation vas developed to approximate numerical
solutions of the Boussinesq equation for one-dimensional, unsteady, unconfined
flow through porous »".«ia (.5). Therefore, the actual hydraulic conductivity
for the drainag? rests was estimated by adjusting its value while solving the
Boussinesq equation until the results matched the measured drainage rates and
saturated depths. The hydraulic conductivity estimates- are summarized in
Appendix A. Determining the hydraulic conductivity in this manner provided
the best estimate obtainable for each test since the Boussinesq solution is
the commonly accepted representation of the actual drainage process. Com-
parisons were made for both steady-state drainage during rainfall and unsteady
drainage following cessation of rainfall.
Comparisons between the Boussinesq solution predictions and the actual
measurements are made in the Appendix B to determine how well the most com-
monly used theoretical representation of the lateral drainage process per-
<	forms. The numerical solution of the Boussinesq equation used in this study
was developed by Skaggs (5). In addition, the Boussinesq solution predictions
are compared with the HELP mcdel predictions in Appendix C. These comparisons
are very important because they defln«t the limits of modeling and demonstrate
how well the HELP equation represents tl.e Boussinesq solution.
COMPARISONS BY HYDRAULIC CONDUCTIVITY
First, the results of the steady-state cases are described. The rainfall
rata was adjusted until the average head on the lintr (which is also the aver-
age depth of saturation, y) reached a_steady-statc height of approximately
12 in. Using this measured value of y and the corresponding measured value of
the steady-state drainage rate, the hydraulic conductivity was computed
explicitly usiag the HELP drainage equation (Equation 1) with the slope and
drainage length employed ia the drainage test. To compute the hydraulic con-
ductivity using the Boussinesq solution, a series of numerical Boussinesq runs
56
-------
were made changing the hydraulic conductivity until the computed steady-state
y having the measured steady-state drainage rate matched the measured value.
A comparison of these values is shown in Table 7, Overall, the computed
hydraulic conductivity using the HELP equation exceeded that using the
Boussinesq solution by an average of 44 percent. Consequently, the HELP sodel
underestimates the drainage rate by 30 percent as predicted by the Boussinesq
solution under steady-state conditions. For fine sand, computed values of
hydraulic conductivity by the HELP equation exceeded the measured valuss using
peraeaaeters by an average of 31 percent; for coarae sand, the average com-
puted value was 10 percent of the measured value.
TABLE ?. COMPARISON OF COMPUTED HYDRAULIC CONDUCmiTirS FOR
STEADY-STATE DRAINAGE
Hydraulic Conductivity
Rainfall
Type of	Slope	Length	Rate	a*	*3**	Tj
Sand
(X)
(ft)
(in./hr)
(cm/sec)
(cm/sec)
(cm/sec.
Fine
2
25.4
0.034
0.008
0,006
0.004
Fine
10
25.4
0.054
0.006
0.004
0.004
fine
10
52.4
0.025
0.006
0.004
0.005
Coarsa
2
25.4
0.071
0.021
0.015
0.220
Coarse
2
52.4
0.033
0.030
0.021
0.220
Coarse
10
25.4
0.169
0.025
0.015
0.220
Coarse
10
52.4
0.105
0.027
0.021
0.220
* Computed hydraulic conductivity using the HELP equation.
** Computed hydraulic conductivity using the numerical Boussinesq solution,
t Hydraulic conductivity measured in laboratory permeaoecers.
For unsteady drainage following cessation ef rainfall, hydraulic con-
ductivity was computed explicitly_using the HELP drainage equation for
instantaneous measured values of y and drainage rate. To compute hydraulic
conductivity using the Boussinesq solution, the following steps were followed:
(a) Unsteady drainage from the physical model was simulated using the numer-
ical Bcussinesq solution with an estimated hydraulic conductivity; (b) at a
given value of y, a correction factor was computed by dividing the measured
drainage rate by the sinulated drainage rate; and 
-------
conductivity. Hydraulic conductivities were computed by these two methods at
various values of y for each experimental case.
Overall, the computed hydraulic conductivity using the HELP tquation
exceeded that using the Boussinesq solution by an average of 13 percent.
Consequently, the HELP equation underestimates the drainage rate by 11 percent
as predicted by the Boussinesq solution under unsteady conditions. These
results were determined using a paired-sample t test to determine whether the
differences in hydraulic conductivity values between the two methods were sta-
tistically different from zero. Table 8 summarizes these results. The test
using all data from all experimental cases concluded that the two methods did
produce hydraulic conductivity values that were statistically different.
Similarly, the tvo methods produced statistically different hydraulic conduc-
tivities using the data for only coarse sand or only fine sand; for only short
model or only long model; and for oi.ly 2-percent slope or 10-percent slope.
However, when the test was conducted for individual experimental cases, the
test concluded that in five out of eight cases, the two methods produced
hydraulic conductivity values chat were not statistically different. The five
cases were: coarse sand in both physical models at 2-percent slope, coarse
sand in the long model at 10-percent 3lcpe, and fine sand in both models at
10-percent slope.
For both steady and unsteady drainage, the hydraulic conductivity com-
puted by the HELP model was shown to exceed that computed by the more rigorous
Bous3inesq solution. Consequently, the HELP model might be expected to under-
estimate the drainage rate from a drainage system where porous media proper-
ties were accurately known. However, in field applications, the magnitude of
this underestimation would probably be much less than the uncertainties in the
in-place porous media properties. This difficulty was highlighted in this
study b> the large differences between laboratory measurements using permease-
ters and apparent in-place hydraulic conductivities. Also, the difference
between the two solution methods was much less for the unsteady case—the case
which generally occurs in field systems.
The commuted hydraulic conductivity of the sand during steady-state
drainage by the Boussinesq solution was about 61 percent of the value during
unsteady drainage. Consequently, if the hydraulic conductivity for unsteady
drainage was used in the HELP model for all types of drainage conditions, the
lateral drainage rate would be underestimated by II percent during unsteady
drainage and overestimated by 14 percent during steady-state drainage.
EFFECTS OF ENTRAPPED AIR
In analyzing the computed hydraulic conductivity values, it wag apparent
that hydraulic conductivity decreased with increasing v. Table 4 shows that
the drainable porosity aLso decreased with increasing y. Lower drainable
porosity could have resulted by two mechanisms: greater air entrapment or
greater compaction. Compaction was 97 percent of maximum throughout the i,amd
layer; therefore, air entrapment must have caused the decrease in drainable
porosity. Lower drainable porosity results in lower cross-sectional area
contributing to drainage; consequently, the hydraulic conductivity decreases.
58
-------
TAHI.K 8 COMI'AKI SON O? COMPUTED HYDRAULIC CONDUCTIVITIES FOR IJNSTKADY DHAINAC.I



Standard


Statist leal

|Jrt C
Mean Value
DevlatIon


i f;n 1 f 1 cance

no • or
of (K„ - K )
of (K - K„)t
ttt
t
1
>( (Ku
" *..)

Data
B ll'
B II
a/2

B
II
Data Croup*
Points**
(cm/sec)
(cm/sec)
Stat If tic
(a ¦= 0.05)
at a -
0.05
All data
93
-0.00309
0.00693
-4.300
-1.960
Not
equa 1
Li> zero
C
43
-0.00589
0.00943
-4.098
-1.963
Not
cqu* 1
to zero
F
50
-0.00608
0.00101
-4. 740
-1 .960
Nor
e<]u;i 1
In zero
2
51
-0.00083
0.00135
-4.375
-1.960
Not
equal
t (i zero
10
30
-0.00584
0.01055
-3.033
-2.045
Not
equa 1
to zero
S
49
-0.00314
0.00815
-2.697
-1 .960
Not
equa 1
to zero
L
44
-0.00303
0.00534
-3.765
-1.960
Not
equal
to zero
C2S
10
-0.00067
0.00107
-1 .981
-2.262
M-« >
equal
zero
C10S
8
-0.01609
0.01491
-J.053
-2.365
Not
equal
to zero
C21.
II
-0.00104
0.00233
-1 .479
-2 .228
May
eqml
zero
C10L
6
-0.00749
0.00746
-2.459
-2.571
May
equal
ze i o
F2S
17
-0.00054
0.0004 7
-4.782
-2. 120
Not
equa 1
to zero
F10S
8
-0.00031
0.J0073
-1.222
-2.365
May
equal
ze ro
F2I.
13
-0.00114
0.00127
-3.235
-2. 179
Not
equa I
to zero
F10I.
8
0.000!?
0.00061
0.552
2.365
May
equa 1
ze ro
* C - coarse aand; S - short model; 2 - 2Z slr.pe.
F ¦ Fine band; I. - long model; 10 = I0Z slope.
*^ Ea jh data point represents one (K - K^) value at particular y.
t Kp - Computed hydraulic conductivity using the numerical Houeulnesq solution,
tt K *= Computed hydraulic conductivity using the IIELI' equation.
H
t -
			 where x	- mean value of (KR - .
s//n	Mu	- hypothetical population mean ~ 0.
8	- standard deviation of (K^ - K(().
n	«=• number of data points.
-------
To determine whether variability in the estimates of the hydraulic con-
ductivity was a function of other variables in addition to average saturated
depth. a series of unequal three-way analysis of variance (ANOVA) tests was
run cm the rseth sets cf estimates. This procedure permits analysis of experi-
ments that have three variables called treatments, which produce variance in
the neasured variable. AKOVA also permits determination of interaction
between variables. The variance »nd the number of duplicates cf treatments
die! net have to be equal in the prot-tdure. The analyses were performed using
Che Statpro software by Penton Software Inc. (6). The test variables include
type of sand, average saturated depth, s»lope, drainage length, rainfall
duration and rainfall intensity. No effects cf rainfall duration and
intensity could be discerned by inspection: therefore, the initial ASOVAs were
run using depth, slope and length as the variables for data sets containing
hydraulic conductivity estimates for one type of sand. These ANDVAs indicated
that the hydraulic conductivity estimates by HELP varied only as a function of
average saturated depth for coarse sand and of average saturated depth and
slops for fin* sand. Running ANCVAs on the estimates generated by the
Boussinesq solution, the estimates varied significantly with depth and slcpe
for both sanir.
No physical reasons are apparent for the variability of the hydraulic
conductivity as a function of slope. Therefore, the variability arises from
inaccuracies in the manners in which the effects of slope are modeled by the
Boussinesq equation and the HELP equation. This difference is presented ia
greater detail later in this section and is Appendices 3 and C.
A regression analysis wis conducted to investigate the relationship
between hydraulic conductivity and y. Evdraulic conductivity, K, was fitted
to a power function of y such that
K - a yfc	0 5)
where values of the regression coefficients a and b are listed in Table 9 for
10 of the experimental cases. This equation applies only to y values ranging
froa 2,5 to JO in. in these drainage tests. In Equation 15, the units for K
are centimeters per second and the units for y are inches. The relationship
between drair.able porcsity and height above the liner is presented in
Section 5.
COMPARISON OF DRAINAGE RATES
To further test the accuracy of the HELP lateral drainage equation, the
relationship between drainage rate and y as predicted by the HELP drainage
equation was compared to measured results for unsteady drainage following ces-
sation of rainfall. The ccaparisons are shown in Figures .25 and 26. Measured
results are represented by the shaded area which defines the range of all mea-
sured data for the given sand, slope and drainage length. Three computed
curves are also plotted. One curve represents the HELP drainage equation
using the hydraulic conductivity measured in the laboratory permeaseters. A
second curve represents the HELP drainage equation using the mean of hydraulic
conductivity estimates obtained froo the Boussinesq solution for all unsteady
drainage Jests performed with the given sand, slope and length. The third
60
-------
TABLE 9. REGRESSION COEFFICIENTS FOR HYDRAULIC CONDUCTIVITY AS A POWER
FUNCTION OF y*
Type of
Sar.d
Slope
Drain
Length
(ft)
a
b
T ine

25,4
0.0951
-0,6650
Fine
10
25.4
0,0243
-0.6166
Fine

52.4
0.067?
-0.7252
Fine
10
52.4
0.0048
0.0964
Coarse
2
25.4
0.1297
-0.6934
Coarse
5
25.4
0.0675
-0.4359
Coarse
10
25.4
0.0477
-0.2392
Coarse
2
52,4
0,1057
-0.5383
Coarse
5
52,4
0.0313
-0.2444
Coarse
10
52,4
0.0395
-0.1896
* The regression equation is K « a y where K - hydraulic conductivity ir.
centimeters per second, y ¦ average depth of saturation or head on the
liner in inches, and a,b - regression coefficients,
curve also represents the HELP equation but uses Equation 15 for hydraulic
conductivity. The curves based on laboratory measurements of hydraulic con-
ductivity resulted it poor fits except for the fev cases where the measured
hydraulic conductivity was ciose to the computed values, "hu again high-
lights the difficulty in .e-j'inating in-place hydraulic conductivity from
laboratory measurements. The curves based on a mean value of hydraulic con-
ductivity fit reasonably well within a _-latively narrow band representing the
y region for which the Bean hydraulic conductivity was computed. Outside this
band, the predicted curves tended to deviate, especially for the short aodel.
The curve b~sed on Equation 15 generally fit well for y values lets than 15 to
20 in. for the fine sand and leas than 12 to 15 in. for the coarse sand.
These also represent the y ranges for which Equation 15 and Table 9 were
derived, Mnra of the predicted curves fit wall at y values close to zero due
to r.he measured capillary water drainage which the HELP equation does not
model.
61
-------
Pradicivd fUwItii
« (it k
O)
• MOftt MOOtl
«% IIOM
AVERaU IICAO (inchtfs)
Pr«dicl*d ffssullai
• N
O fttOT I
Ui
~—
or
lONO MOOCi
<
rr
D
AVEHAGl Hi AO (melius)
Pr«dKi») K«iu|ii4
* lab K
O fatf. M
cn
10% tlO'I
i; rr j
AVIKAGI IK AO (locties)
A-i	
Predicted Riiullu
• lab K
li Sweat H
I
15-
£
•<
a:
lONO MOOfl
to% llOM
oc
n
1/ ~l6 Si
AVIRALC lit AO (inches)
Figure 25.
heasured drainage rate following rainfall vs. average head niiuve the liner compared
to HELP drainage equation predictions for t'lne sand In both physical models at
slopes of 2 and 10 percent.
-------
AVERAGE ItfAU (inLhwb)
AVERAtl I*AD (inchas)
AVLRAW ItLACI (inches)
AVEKACl Ml All (inches!
Figure 26.
Measured drainage rate following rainfall vs. average h*ad above the liner compared to
HELP equation predictions for coarse sand In both physical models at slopes of 2 and
10 percent.
-------
SIMULATIONS ST REL? MODEL
Figures 27-30 shew hew veil th
-------
?« !-•m itAiM ai ii 14 ttt tm
SHQ8I wnotl A! 21 UP|
I
-c
a
M
( D M Mill *1 0. 57 IN /IN
MUHI MUX. *1 131 a«
.a
fc?
£
3
i lo IT la k <•
IMC Qio>j"s!
a*
VJ>
?« !-**> KAN *» U H It /m
W0*f «am »r /i m»*c
mimiiu
a AC IIJU
X
I
(. 0 M Ml* il 1:1 m /M
SMCR* MML *1 10. si arc
"Uo°o?L"r-~—
"••to o
II-
y I p.ure 27, Htreurtd drainage rate and average sa' urat pd depth vs. tin' compared to IIKI.." j/rt* ill r». ion
for fine sand In the aliort model at slopes of 2 and 10 percent.
-------
1.2 ** *A|M *1 0 IN, /MM
ICNC	A! (Ill SiUPf
- mcoicicfi
t* ACIU41
8
I' .
4.*m iuim ai b.si m Am
itxr> MOQii *i n Siort
- PMLOIC 1(0
o KIUM.
.12

§
-------
i i «: RMM ai (i ^ in a«
fcOfl «£i! *1 Idl S»tft
- PHU#uiia
O A« IUAL
i
a>
3
V la h It*
I U-m RAIN A? 0 *• l*/Ht
Mim ai ?i stcrt
- mmuio
a ACruAl
j.
«
ta
la
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n
Figure
8 o hi rain hi a. si in /iti
SiiCRT nnBft it MOPt
PWOIUIQ
KIUAI
'*V^
1r-»
£ £
II* 
-------
2 m RAlH *1 0 57 IN />H
LUNC M0UT1 Al IQ1 SI
- PNtOICTIO
u ACTUAL
ft RAIN At 0 U IN /M
I QNt MUCH.I Al 21 SLGPl
i
Ui
or
z
•<
S
0-

00
6 I-KM RAIN Al 0 » IN /Mt
10NG WOfL Af ?X SLOPt
— pwntcrio
o AC1U4L

^-4-1

Jo	2a
TlHo (hta**c)
5
£
&
t 
-------
.25
6.2-WR SAIN AT 0.54 IN. ."Hfl
— PSEDICTED
a ACTUAL
!
m
UJ
=c
S
B
.05-
6, 2-HR SAIN AT 0. 54 IH. Ati
Ui
I
c
a.
a
10-1
TIME (hours)
Figure 31. Measured drainage race and average saturated depth vs.
time conpared to HELP prediction for fine sand in the
long model at i0-percent slope using oean values for
hydraulic conductivity and drainable porosity.
69
-------
& 1-HR BAIN AT 0.56 IK/m
m
Q_
en
.3-
z
8
TIME (hours)
6, l-W RAIN A? 0.56 IN./MR
UJ
at
Figure 32. Measured drainage race and average saturated depth va.
tia« compared to HELP prediction for coarse aand in the
short model ac 10-percent slope using mean values for
hydraulic conductivity and drainable porosity.
70
-------
created as duplicates in this analysis. First, th« effect of drainage length
was isolated by using Equation 1 to compute the ratio of drairage rata for the
long model (Q^) to drainage rate for the short model (Q ) given the sane
drainage media and constant values for a and y, This fitio was tultipli-'d by
the measured Qg to predict Q, , The difference betveen measured and predicted
was assumed to be due to a combination of experimental error and simplify-
ing assumptions used in the derivation of the HELP drainage e^uatios.
Table 10 summarizes these comparisons, and Figure 33 plots the predicted
versus measured Q^. A regression analysis was conducted using Che data in
Table 10 and Figure 33 to find the slope of the best-fit line vhich also
passed through the origin. The regression analyses performed to evaluate the
HELP equation are summarized in^Table 11. The resulting slope was 0.82 with a
coefficient of 1#terminal!ion (r ) of 0.99. The 95-percent confidence interval
for the slope did not include the value of 1.00. The slope should equal 1.00
and Che intercept should equal 0.0 if the predicted and actual results are
identical and the predictive method is accurate. Therefore, Qf as measured in
the unsteady drainage tests and as predicted by the HELP equation based on a
measured Q are statistically different. The HELP lateral drainage equation
overestimates the decrease in drainage rate resulting from an increase in
length given the same sand, slope, and y.
The effect of slope was similarly isolated by using Equation 1 to compute
the ratio of drainage rate for the 10-percent slope (Qirj?) to the drainage
race for the 2-percent slope (Oj*) given the sane sand and constant values for
L and y. This ratio was multiplied by the measured Q,. to predict Q,.
Table 12 summarises these comparisons, and Figure 34 plots the predicted Q,a»
versus measured Q-0_. A regression analysis was conducted using the data In"
Table 12 and Figure 34 to find the slope of Che best-fit line wtjich also
passed through the origin. The resulting slope was 1.30 with r* - 0.98. The
95-pircer.t confidence interval for Che slope of this li**e did not Include the
value of 1.0C. Therefore, Q^q« as measured in the laboratory and as predicted
by the HELP equaLion based on a measured are statistically different. The
HELP lateral drainage equation overestimates the increase ia drainage rate
resulting from an increase in slope.
Similarlyi Equation 1 wus used to compute the ratio of drainage rate for
y • 12 in. CQI?„) to drainage rati fer y - 6 in. (Q^») given the sane sand ar.d
conscar.t values for L and i. In the calculation of this ratio it was assumed
Chat the hydraulic conductivity at y - 12 in. was equal Co 0.732 times the
hydraulic conductivity at y • 6 in. based on Equation 15. This ratio was
multiplied by the measured Q,,, to predict Q, . Table 13 summarizes these
comparisons, and Figure 35 plots the predic?ed Q.," versus neasured Q ,
T|e slope of the best-fit line passing through trie origin was 1.02 with
r - 0.99. The 95-percent confidence interval for the slope of this line
included the value 1.00. Therefore, Q^" as measured in the laboratory and as
predicted by the HELP equation based on a measured Qg„ are not statistically
different. The HELP lateral drainage equation accounts for the effect of
changes in y very well.
In the HELP lateral drainage equation (Equation L) the slope (a) and
drainage length (L) occur as a product several times. This product physically
represents the Head above the drain contributed by Che slope of the liner.
71
-------
TABLE 10. EFFECT OF LENGTH IN THE HEL? DRAINAGE EQUATION*
Q./Q	Measured Measured Predicted**
Type of	y	Slope frots H§L?	Q	Qt	Q.
Sand	(la.)	(I) Ecuador,	(gpg)	(gpfl)	(gpfi)
Fine 6
2
0
737
0
030
0.025
0.022
8
2
0
689
0
044
0.030
0.030

5
0
868
0
043
0.040
0.037

10
1
005
0
058
0.061
0.058
12
2
0
626
0
056
0.039
0.035

5
0
802
0
058
0.051
0.047

10
0
956
0
072
0.070
0.069
14
2
0
604
0
067
0.045
0.040

5
0
775
0
068
0.057
0.053

10
0
933
0
082
0.081
0.077
Coarse 6
2
0
737
0
056
0.052
0,041

5
0
909
0
098
0.094
Q .089

10
1
033
0
164
0.166
0.169
8
2
0
689
0
070
0.067
0.048

5
0
868
. 0
1.-3
0.116
0.107

10
1
005
0
183
0.204
0.184
12
2
0
626
0
119
0.106
0.074

j
0
802
0
177
0,169
0.142

10
0
936
0
244
0.287
0.233
14
2
0
€04
0
151
0.139
0.091

5
0.
775
0
218
0.210
0.169

10
0.
933
0
264
0.334
0.246
* ¦ Drainage rate froa long model.
Q - Drainage rata froa short model,
s
*+ Prtdictpd Q, ¦ (Q, /Q frum HELP Equation) x (Measured Q }.
L	L S	S
72
-------
.25-
15"
. 1
^ FINE SAND, y
a FINE SAND, y
FINE SAND, y
COARSE SAND.
COARSE SANU.
COARSE
COARSE SAND,
1	T
.1	.15	.2
ACTUAL DRAINAGE RATE
Figure 33. Predicted vs. actual drainage rate UBlng measured drainage from short model
to predict drainage from long model.
-------
TABLE 11. REGRESSION ANALYSES SUMMARY FOR EVALUATION
OF THE SEU1 LATERAL DRAINAGE EQUATION
952 Conftderce 1laics
Data Set
0 /C *
V \v
Vqa
Intercept
N**
2
r
Effect of length,
constant a
0.819
0.749 to
0.890
-0.0070 to
0.0125
22
0.987
Effect of s^ooe
1.302
1.086 to
1.519
-0.0269 to
0.0511
15
0.979
Effect of y
1.023
0.866 to
1.193
-0.0207 to
0.0279
10
0.989
Effect nf aL,
cons taur L
1.190
1.115 to
1.276
-0.0099 to
0.0130
15
0.996
* Qu - Drainage race predicted by HEL? equation.
Q" - Drainage race measured in drainage test.
** fj* « number of values In data set.
Therefore, the effect of aL was similarly Isolated by using Equation 1 to com-
pute the rzzlo of drainage rate cor an aL value of about 30.5 in. (Q,n ,«,) t.o
drainage rate for an aL value of about 15.3 in. (Qic ¦,.¦) » given the same 3and
and constant values of L and 7. This ratio was multiplied by tha asasursf
Q.j t0 predict j,.. Table 14 summarizes these comparisons and Figure 36
plots the predicted	versus aeasured Q.^	A regression analysi-j was
conducted using the data in Table 14 and Figure 36 to find the slupc ?£ Che
best-fit line which also passed through the origin. The resuliing slopi
1.20 with r • 0.996. The 95-percent confidence interval for the slo>r did
not include the value of 1.00. Therefore, Q-- as measured in the labora-
tory and as predicted by th*. HELP equaticn bastd on a measured Q . are sta-
tist ically different. The HEL? lateral drain;3e equation overestiaatis :ha
increase in drainage rate resulting from an increase in slope.
Earlier, the effect of length was isolated frota the effects of slope,
type of slope, type of sand, and y, but it was not isolated from the effect of
aL. To isolate cbe effects of length from the effect of aL, the rauio of
drainage rate for the long model (0r) to the drainage rar.e for the short model
(Q ) given the same sand and constant values of y and aL was computed using
Equation I. This ratio wis multiplied by the aeasured Q to predict 0.,
Table 15 summarizes these comparisons and Figure 37 plot!' the predicted Q
versus measured Q . A regression analysis was conducted using the data in
Table 15 and Figufe 37 to 
-------
TA3LE 12. EFTECT Of SLOPE IN THE HELP DRAINAGE EQUATION*
Type of
Sand
L
(In.)
V
(in.
^ 10Z ^2Z
from EELP
) Equation
Measured
q2Z
(gpta)
Measured
Q10X
(g?a>)
Predicted**
°102
(*pa)
F ine
305.4
6
3.187
0.030
0.050
0.096


8
2.804
0.044
0 .058
0.123


12
2.300
0.056
0.072
0.129


14
2.12b
0.067
0.082
0.143

629.4
8
4.090
0.030
0.061
0.123


12
3.512
0.039
0.070
0.137


14
3.287
0.045
0.081
0.148
Coarse
305.4
6
3.187
0.056
0.164
0,178


8
2.804
0.070
0.183
0.196


12
2.300
0.119
0.244
0.274


14
2.128
0.151
0.264
0.321

629.4
6
4 a 464
0.052
0.166
0.232


8
4.090
0.067
. 0.204
0.274


12
3.512
0.106
0.287
0.372


14
3.287
0.129
0.334
0.457
* 0 -
? Y
Q m
yioz
Drainage
Drainage
race
race
from 2-percent
from 10-percent
slope.
slope.


** Predict
ad Q10% *
CQlOj'/Qi« fr0D HELP
Equation)
x (Measured
"zx5'
75
-------
E
C_
cn
c
Cd
LD
¦C
en
a
CD
<_>
a
LlJ
cr
Q_
305
305
305.4
305.4
629. 4
629.4
629. 4
629.4'
.08
ACTUAL
. 12
DRAINAGE
Figure 34
Predicted vs. actual drainage rate using measured dralnape at 2-percent slope
to predict drainage at 10-percent slope.
-------
TABLE 13. EFFECT OF AVFRAGE DEPTH OF SATURATION IN THE HELP
DRAINAGE EQUATION*



Q, Vi/Qfi'i
Measured
Measured
Predicted**
Type of
L
Slooe
from HELP
Q6"
Q.

Ql2"
Sand
(in.)
r.)
Equation
(gp®)
(gpm)
(Spn)
Fine
305.4
i.
2.357
0.030
0
056
0.071


5
1.918
0.032
0
058
0.062


10
1.693
0.050
0
c;2
0.085

529.4
*5
1.99S
0.025
0
039
0.050
Coarse
303 .4
2
k .357
0.056
0
119
0.132


r
1.918
0.098 .
0
177
0.188


10
1.698
0.164
0
244
0.279

679.4
2
1.998
0.052
0
106
0.104


5
1.695
0.094
0
168
0.160


to
1.574
0. 166
0
237
0.261
* Q^., - Drainage race f>om v » -> in.
Qt1«i " Drainage rate from v » i 2 in,
** Predicted Q,»„ ¦ (C, -»«./Q .. from HELP ciuacion) x '.Measured
I c	I i. 0	h
77
-------
E
CL
cn
Ck
L = 3()rj. 4'
, « -= 77.
~
L - 305.4'
. « - M
o
1 - 305.4'
. a = m
A
L - 629.4'
.. « - 21
¦
L = 629. 4'
. a = b I
•
L = 629.4'
. «= m
¦c
CK
CD
¦C
¦c
an
CD
n
(_j
a
U)
oc
Q_
T
1	.2
ACTUAL DRAINAGE-: RATE (gpm)
Figure 35. Predicted vs. actual dralnag. rate using measured drainage «t y - 6 in.
to predict drainage at y * 12 In.
-------
TABLE 14. EFFECT OF aL IN THE HELP DRAINAGE EQUATION*
Type of L
Sand (ia.)
y
(ia.)
Q30.5"/Q15.3"
from HELP
Equation
Measured
Q15.3"
(g?ns)
Measured Predicted**
Q30.5" ^30,5"
(Spm) (gpn)
Fine 305,4
6
1.815
0.032
0
050
0.058

8
1.736
0.043
0
058
0.075

12
1.609
Q. 058
0
072
0.093

14
1.557
0.36<3
0
082
0.106
629.4
3
1.736
0.034
0
039
0.059

12
1.609
0.043
0
050
0.069

14
1.557
0.049
0
056
0.076
Coarse 305.4
6
1.815
0.098
0
164
0.178

8
1.736
0.123
0
183
0.213

12
1.609
0.177
0
244
0.284

14
1.557
0.218
0
264
0 339
629.4
6
1.815
0.060
0
091
f .109

8
1.736
0.076
0
113
0.132

12
1.609
0.117
0
163
O.188

14
1.557
0.152
0
205
0.237
* Q,0 • Drainage
Qj2'3» " Drainage
rate
rate
for aL • 30.5 in.
for aL - 15.3 ia.




** Predicted ¦
CQ30
5"^L5 3" ^ro® Equation)
x (Measured
Qjj 3«).
79
-------
CD
o
E
d_
CD
Ul
Q1
11J
LD
<
>—*
<
cr.
tn
en
UJ
»- -
CD
iii
l YL
n_
305.4
305.4
305.4
305.4
629.4
G In.
15.1 In.
-------
TABLE 15. EFFECT OF LENGTH IN THE 3EL? DRAINAGE EQUATION GIVEN
CONSTAiVT sL*
Q, >'Q.
Measured

Predicted**
Type of
Sand
aL
(in.)
V
(in.)
:• ~~ 3
\ .i_an
9s
(*po)
ql
(g?m)
Q,
L,
(gpm)
Fine
15.1
6
0.485
0
032
0.U29
0.016
Fine
15.3
3
0.485
0
043
Q.C34
0.021
Fine
15.3
12
C.485
0
058
0.043
0.028
Fine
15.3
14
0.if.5
c
068
0.049
0.023
Fine
30. S
8
0.485
0
053
3.039
0.028
Fine
30.5
12
0.-35
0
0"2
Q.05C
0.035
F ine
3C.5
14
0.435
0
062
0.056
0.040
Coarse
15.3
6
0.485
c
098
0.060
0.048
Coarse
15.3
8
0.485
0
123
0.076
0 060
Coarse
15.3
12
0.485
0
177
0.117
C .086
Coarse
15.3
14
0.485
0
218
0.152
0.106
Coarse
30.5
£
0,i85
0
164
0.091
0.030
Coarse
30.5
8
0.485
c
183
0.113
0.039
Coarse
30.5
12
0.485
0
244
0.163
0.113
Coarse
30.5
14
0.485
0
264
0.205
0.128
* Qj_ " Drainage rate from Long model.
Q » Drainage race from shore model,
s
** Predicted Q, ¦ (Q. /Q from HELP equation) ;< (Measured Q ).
L	L 3	S
81
-------
«L
«L - 15.3
"L = 15.3
«L = 3D. 5
«L = 30.5
«L = 30.5
«L = 30.5
E
Q_
CD
.15"
Lul
LU
CD
¦C
~z.
I—I
a
LU
5 .05-
a
LxJ
Ctl
Q_
.05
ACTUAL DRAINAGE RATE (gpm)
Figure 37.
Predicted vs. actual drainage rate using measured drainage from short
to predict drainage from long model having the same value of al..
-------
SECTION 7
SUMMARY AND CONCLUSIONS
Drainage testa were performed on cvo large-scale physical models of land-
fill liner/drain systems Co examine Che effeces thac che length, slope and
hydraulic conductivity of the drain layer and the depth of saturation above
the liaer have on Che subsurface lateral drainage rate. The models have dif-
ferent lengths. 25.4 ft and 52,4 ft, and adjustable slope rangirg from 2 to 10
percent The models were filled wich a 3-ft sand drain layer overlying a l-ft
clay liner, A 2-in. layer of gravel was placed under Che liner to collect
seepage from the clay liner, Drainage from tvo different sands was examined
it, both models.
•
Several drainage tests wer? run on each configuration of the models by
applying water as r?infall Co che surfacn of the sand layer and Chen measuring
the water cable along the lengch of the models and Che lateral drainage volume
as a function of time. Lateral drainage and water table profiles were mea-
sured during periods of increasing, decreasing, and stetdy-stace drainage
rates. In total, more Chan 60 cescs were performed.
The hydraulic conductivity of the tand was measured in the laboratory
permeaaecers, but its value was apparently quite different and varied in the
drainage testa. The hydraulic conducel\ity of the sand ac various depths cf
saturation was estimated for each test vsing the Bousslnesq solution of
Darcy's lav for unsteady, unconfined flew through porous media and the HELP
lateral drainage equation, These hydraulic conductivity values were compared
to deter.nine the agreement between the HELP model and Che Bousslnesq solution.
For steady-state drainage the HELP modtl estimates of the hydraulic ccnuuc*
Civity were 44 perenne greater than Che Boussinesq solution estimates, Th'.j
result means that the HELP model underestimated che steady-state latera1
drainage rata predicted by the Bousslnesq solution by 30 percent. The HELP
estimates were 31 percent greater than the laboratory measurements for the
fine sand and 90 percent less than the laboratory measurements for the coarse
sand. For unsteady drainage the HEL? model estimates were only 13 percent
greater Chan the Boussir.ssq solution estimates, which would underpredict the
lateral drainage rate by 11 percent. The underpreJiction of the cumulative
latetai drainage volume would be expected to be very small since che removal
rate of vater from the drain layer by all other means is much smaller than the
lateral drainage rata. Consequently, the effect of differences in the pre-
dicted and actual drainage Clmea i« small.
The differences between the laboratory measurement of the hydraulic con-
ductivity and either of the tvo estimates were much larger than the differ-
ences between Che estlaaCes. In addition, the hydraulic conductivity varied
83
-------
— _	^ wae uejjcn or saturation, apparently due to entrapment of air
in the sand. Similarly, the dralnable porosity varied as a function of depth
of saturation due to the air. These phenomena cake it very difficult to model
the lateral drainage process and produce good agreement becveen the predicted
and actual results for drainage rate and depth of saturation as a function of
t i line .
An analysis was performed to determine how weJ1 the lateral drainage
equation in the HELP modal accounts for the effects of drainage length, slope
of the liner, average depth of saturation and head above the drain contributed
by the liner (aL) in the estimation of the drainage rftte. The drainage equa-
tion overestimates the decrease in drainage rate by an increare in length
given the same sand, slope, depth of saturation and >ead £.lov# the drain,
Similarly, the equation overestimated the increase in drainage rate resulting
from an increase in slope and head above the drain. The effects of dept'. of
saturation predicted by the equation agreed very well with the actual results.
The following conclusions and recommendations are made. Lateral drainage
in landfill liner/drain systems is quite variable, probably due to air entrap-
ment. The hydraulic conductivity measurement made in the laboratory is quite
different than the in-place value. Consequently, the estimation of the lat-
eral drainage race is prone to considerable error despite having a good equa-
tion or solution method for the estimation. Nevertheless, the prediction of
the cumulative volume of lateral drainage is likely to be quite good since the
depth of saturation will be overpredlcted if the drainage rare is underpre-
dicted and vice versa, thereby adjusting the drainage rate. However, the pre-
dicted deprh of saturation will be quite different from the measured value.
The lateral drainage equation in the HELP model performs very well for
tests on models at 2-percent slope but overpredlcted drainage and depths at
10-percent slope. The model overestimates the effect of drainage length in
reducing the drainage rate and also overestimates the effects of slope and
head above the drain in increasing the drainags rate. Therefore, additional
refinement in the equation should be performed to make the equation valid over
a wider range of slopes and drainage lengths. Nevertheless, the HELP equation
does provide a good estimation of the lateral drainage volume.
Evaluation of the effects of drainage length, slope of the liner, depth
of saturation ar.d head above the drain on the drainage rate predicted by the
Boussinesq solution should be performed to determine whether the effects
observed with the FEFLP drainage equation are unique. Similarly, an additional
data set of drainage results should be collected to determine vhether the
effects are unique to this data set. Additional data should be collected for
longer drainage lengths and greater slopes and from actual landfill/liner
systems.
84
-------
REFERENCES
1.	Schroeder, P. R.» Gibson, A, C.. and Ssclen, M. D, The Hydrologic
Evaluation of Landfill Performance (HELP) Model, Voluae II. Documentation
for Version I. E?A/530-SV-€4-0l0, U.S. Environmental Protection Agency,
Offict of Solid Waste and Emergency Response, Washington, D.C, 1984,
256 pp.
2.	Schroeder, P. R.» Morgan, J. M,, Wtlski, T. M., and Gibson, A. C, The
Hydrologic Evaluation of Laadfill Performance (HELP) Model. Voluae I.
User's Guide for Version I. EPA/530-SV-S4-C09, U.S. Environaetita1
Protection Agency, Office of Solid Waste and Emergency Response,
Washington, D.C., 1984. 120 pp.
3.	U.S. Environmental Protection Agency. RCRA Guidance Document for Landfill
Design. Federil Register, General Services Administration, Washington,
D.C., July 26, 1982.
4.	Headquarters, Departnent of the Army, Office, Chief of Engineers. Labora-
tory Soils Testing. Engineer Manu/.l H10-2-1906, Washington, D.C., 1970.
5.	Skaggs, R. W. Modification to DRAINMOD to Consider Drainage from and
Seepage Through a Landfill, Appendix A. N'uoerical Solution to the
Boussinesq Equation for Transient Water Movement over an Inclined
Restrictive Layer, Draft Report, U.S. Environmental Protection Agency,
Cincinnati, OH, 1982. 15 pp.
6.	Penton Software, Inc. Stacpro User's Manual. Sew York, 1965,
85
-------
BIBLIOGRAPHY
Baver, I. D.» Waiter K, Gardner, and W'ilfcrd R. Gardner. 1972. Soil Physics.
John Wiley and Sons, New York. 484 pp.
Carlson, E. J. 1971. Drainage from Level and Sloping Land. REC-EXC-71-44,
Engineering and Research Center, Bureau of Reclamation, Denver, CO.
43 pp.
Dak.i, J. K. K. 1972. Essentials of Engineering Hydraulics. John Wiley and
Sons, Inc., New York. 392 pp.
Freize, 1. All o, anJ John A. Cherry. 1979. Groundwater. Prentice-Hall,
Inc., Er.glevood Cliffs, HJ. 553 pp.
Hillel, Daniel. 1971. Soil -,nd Kate.': Physical Principles and Processes.
Acadenic Press, New York. 288 pp.
Hillel, Daniel. 1982. Introduction to Soil Physics. Acadeaic Press,
Mew York. 364 pp.
List, E. John. 1965. Steady Flow to Tile drains Above an Impervious Layer -
A Theoretical Study. KH-R-9, California Institute of Technology,
Pasadena, CA, 60 pp.
Luthin, Janes N. 1966, Drainage Engineering. John Wiley and Sons, Inc.,
New York. 250 pp,
86
-------
APPENDIX A
HYDRAULIC CONDUCTIVITY ESTIMATES
The analysis performed la Section 6 of this report required estimates of
th* hydraulic conductivity of the drainage media. These estimates were
generated using the ouaerical Boussiaesq solution and the EEL? oodel lateral
drainage equation. Estimates were generated for sost of the drainage tests
incindiug steady-state drainage, unsteady drainage following rainfall, and
unsteady drainage from presaturated sand. The hydraulic conductivity values
were astimated at several depths for the unsteady drainage tests. Values for
steady-state drainage, unsteady drainage following rainfall, and unsteady
drainage from presaturated sand are given, respectively, in Tables A-l, A-2,
and A-3.
TABLE A-2. HYDRAULIC CONDUCTIVITY ESTIMATES FOR STEADY-STATE
DRAINAGE
Model	Average Drain Hydraulic Conductivity
Type	Length Slope	Depth	Rate	(in./hr)
of Sand
(in.)
(X)
(in.)
(in./hr)
Boussinesq
HELP
Pine
205.4
1.8
12.6
0.0336
0.0061
0.0084
Fine
305.4
10.4
13.2
0.9543
C.0Cj7
0.0055
Fine
629.4
10.3
12.?
0.0249
0.0041
0.0058
Coarse
305.4
1.9
11.4
0.C708
0.015
0.021
Ccarse
305.4
10.0
11.3
0.1688
0.015
0.025
Coarse
629.4
2.3
11.3
0.0330
0.021
0.030
Coarse
629.4
9.9
11.8
0.1050
0.021
0.027
87
-------
TABLE A-2. HYDRAULIC CONDUCTIVITY ESTIMATES FOR ¦JSSTfAD'?
DRAINAGE FOLLOWING RAINFALL

Model

Rainfall

Kydraulic Conduc
tivity
Type
Length
Slope
l>jration
Depth
(in./hr)

f Sand
(in.)
C)
(hr)
(in.)
Scussines:
UCT p
Fine
305.4
1.8
1
9
C.0138
0.0147
Fine
305.4
1.8
1
8
0.0151
o.oisa
Fine
305.4
1.8
1
7
0.0173
0.0180
Fine
305.4
1.8
2
19
0.0048
0.0063
Fine
305.4
1.8
2
17
0.01 11
0.0107
Fine
305.4
1.8
2
10
0.0200
0.0211
Fine
305.4
1.8
2
10
0.0172
0.0180
Fine
305.4
1.8
6
17
0.0116
0.0117
Fine
305.4
l.S
6
9
0.0182
0.0183
Fine
305.4
1.8
24
19
0,0064
0.0063
Fine
305.4
1.3
24
15
0.0078
0.0080
Fine
305.4
1.8
24
5
0.0212
0.0218
Fine
305.4
:.8
24
4
0.0251
0.0262
Fine
305.4
4.9
6
P
0.0061
0,0069
Fine
305.4
4.9
6
7
0.0106
0.0112
Fine
305.4
10.4
1
6
0.0083
0.0C96
fine
305.4
10.4
1
5
0.0092
J.0109
Fine
305.4
10.4
2
17
O.OC61
0.0059
Fine
305.4
10.4
2
11
0.0067
0.0066
Fine
305.4
10,4
6
16
0.0060
0.0058
Fine
305.4
10.4
6
8
0.0079
0.0082
Fine
305.4
10.4
24
15
0.0039
0.0033
Fine
i05.4
10.4
24
12
0.0044
0.0043
Fine
629.4
2.1
1
8
0.0164
0.0165
Fine
629.4
2.1
1
7
0.0179
0.0184
Fine
629.4
2.1
2
20
0.0070
0.0108
Fine
629.4
2.1
2
16
0.0086
0.0104
Fine
629.4
2.1
2
10
0.0132
0.0135
Fine
629.4
7.1
6
20
0.0084
0.0099
Fine
629.4
2.1
6
10
0.0134
0.0136
(Continued)
88
-------
TABLE A-2. (CONTINUED)
Type
of Sand
Model
Length
fin,)
Slope
'25
Rainfall
Duration
Chr)
Depth
(in.)
Hydraulic Conductivity
(in,/hr)
Boussinasa HEL3
Fine
629.4
2.1
24
19
0,0081
0.0092
Fiat
629.4
2. 1
24
9
0.0128
0.0132
Fin«
629.4
5.0
6
ia
0.0050
0.0075
Fine
629.4
5.0
6
9
0.0073
0.0107
Fine
629.4
10.3
1
13
0.0054
0.0062
Flo*
629.4
10.3
1
11
0.0062
0.0069
Fi»«
629.4
10.3
2
17
0.0061
0.0058
Fine
629.4
1C.3
2
14
0.0057
0.0055
?ine
629.4
10.3
6
36
0.0068
0.00'il
Fin*
629.4
10.3
6
11
0.0064
0.00(3
Fine
629.4
10.3
24
1?
0.0C65
0.0C58
Fine
629.4
10.3
24
13
0.0062
0.0058
Coarse
305.4
1.9
2
15
0.0228
0.0218
Coarse
305.4
1.9
2
10
0.0276
0.0274
Coarse
305.-
1.9
2
4
0.0720
0.07*8
Cv-arse
305.4
1.9
0
15
0,0225
U.0227
Coarse
305.4
1.9
6
11
0.0279
0.0274
Coarse
305.4
1.9
6
6
0.0413
0.0431
Coarse
305.4
1.9
24
8
0.0317
0.0319
Coarse
30i. 4
1.9
24
8
0.0316
0.0315
Coirse
305.4
1.9
24
5
0.0508
0.052?
Coarse
305.4
1.9
24
4
0.0734
0.0745
Coarse
305.4
5.4
2
9
0.0309
0.0306
Coarse
305.4
5.4
2
5
0.0383
0.0409
Coarse
305.4
5.4
6
9
0.0253
0.0251
Coarse
3015.4
5.4
6
4
0.0394
0.0418
Coarse
305.4
10.0
2
6
0.0348
0.0370
Ccars*
305.4
10.0
2
2
0.0472
0.0790
Coarse
303,4
10.0
6
7
0.0300
0.0334
Coarse
305.4
10.0
6
2
0.0456
0.0747
(Continued)
89
-------
TABLE A-2. (CONCLUDED)
Type
of Sand
Model
Length
(In.)
Slope
(?)
Rainfall
Duration
(hr)
Depth
(in.)
Hydrav.lic Conductivity
(in./hr)
Bcussinesc HEL?
Coarse
305.4
10.0
24
3
0.0301
0. Oil 24
Coarse
305.4
10.0
24
1
0.0431
0.0814
Coarse
6:9.4
2.3
2
16
0.0292
0.0238
Coarse
629.4
2.3
2
11
0.0301
0.0294
Coarse
629.4
2.3
2
5 •
0.0500
0.05C8
Coarse
629.4
2.3
6
11
0.0277
0.0276
Coarse
629.4
2.3
6
3
0.0692
0,0761
Coarse
629.4
2.3
:4
8
0.0354
0.0350
Coarse
629.4
2.3 >
24
£
0.0553
0.0582
Coarse
629.4
5.2
2
i:
0,0213
C.L 294
Coa-se
629.4
5.2
2
4
0.0281
C.0-.63
Coats*
629.4
5.2
6
10
0.015?
G.02S'
Coarye
629.4
5-2
e
5
0.0164
0.0349
Coarsa
629.4
9.9
2
9
0.0278
0.0290
Coarse
629.4
9.9
2
7
0.0267
0.0309
Coarse
629.4
9.9
6
9
0.0282
0.0300
Coarse
629.4
9.9
6
4
0.0362
0.0449
Coarse
629.4
9.9
24
6
0.0243
0.0325
Coarse
629.4
9.9
24
3
0.0313
0.0527
90
-------
TABLE A-3. HTDRAULIC CONDUCTIVITY ESTIMATES FOR UNSTEADY
DRAINAGE FSOM PRXSATURATED SAND
Type
of Sand
Model
Length
(ii.)
Slope
(I)
Depth
(in.)
Hydraulic
(in
Bousslntsq
Conductivity
. /hr)
HELP
Fine
305.4
1.8
14
0.0150
0.0156
"ine
305.4
1.3
12
0.0111
0.0110
Fine
3C5.4
1.8
10
0.013"!
0.0191
Fine
305.4
1.8
9
0.0147
0.0148
Flea
629.4
2.1
24
0.0079
0.0116
Fine
629.4
2.1
10
0.0155
0.0161
Fink
629.4
2.1
6
0.0201
0.0201
Fine
629.4
2.1
4
0.0238
0.0245
Coarse
305.4
1C.0
10
0.0243
0.0244
Coarse
305.4
10.0
5
0.0319
0.0434
Coarse
629.4
2.3
13
0.0260
0.0257
Coarse
629.4
2.3
7
0.0446
0.0454
Coarse
629.4
1.3
6
0.0456
0.0451
Coarse
629.4
2,3
3
0.0621
0.0647
91
-------
APPENDIX 3
COMPARISON! BETVEIN BOUSSINESQ SOLUTION
AND HELP LATERAL DRAINAGE EQUATION
The HELP lateral drainage equation was developed from numerical solutions
of the Boussinesq equation for saturated unconfined lateral flow through
porous aedia under unsteady drainage conditions. Consequently, the tvo
methods should predict the same drainage rate under the same unsteady drainage
conditions when the same values art used for the design parameters. ihia
appendix compares predictions by the tvo aethods using actual test data to
deterxiae how well the HELP equation represents the Boussinesq solution.
In Section 6 hydraulic conductivities predicted by the tvo methods were
briefly compared. Since the drainage rate computation is directly propor-
tional to the hydraulic conductivity value in both aethods, Che estimates of
hydraulic conductivity using measured values of drainage rata, depth of satu-
ration, drainage length, jlope end infiltration rate can be directly coapared
to deteraine the agreement between the methodw. Comparisons are made la this
appendix for steady-state drainage, unsteady drainage following rainfall and
uncteady drainage from presaturated sand using the hydraulic conductivity
values presented in Appendix A. Comparisons are aljo made to determine th*
effect.1? of slope, drainage length, type of sand and depth of saturation on the
agreement between the tvo equations.
Steady-state drainage differs from unsteady drainage since recharge
alters the profile of the depth of saturation as a function of distance from
the drain. Consequently, the drainage rate at a given average depth of satu-
ration la greater for steady-state drainage than for unsteady drainage whtn
the profile is falling, and is less than when the profile is rising. In
actual drainage tests the amount of difference is confounded by the difference
in air entrapment between the tvo modes of drainage. Comparisons between pre-
dicted hydraulic conductivity values for steady-state and unsteady drainage by
the numerical solution of the Boussinesq equttiou and the HELP lateral drain-
age equation are summarized in Table B-L where the ratio of hydraulic conduc-
tivity values is presented. The ratio of hydraulic conductivity estimates for
steady-state drainage to those for unsteady drainage was 0.614 using the Bous-
sinesq solution and 0,831 using the HELP equation. Therefore, using a con-
stant hydraulic conductivity value for all rypes of drainage would produce
greater error with the Bouaslaeirq solution than with the HELP equation. These
tvo regressions are presented in Figure B-l. A ratio of I.000 and aa inter-
cept of 0.00C0 indicate that the data are identical. In all of th-s regres-
sions presented in this appendix, the intercept was not significantly
different from zero. Therefore, the regressions were run using an Intercept
of 0.0000 to deteraine the ratios, K1/K2, presented in Tables B-l and B-2.
52
-------
TABLE 3-1. BTDRA'JIIC CONDUCT! 7ITY REGRESSION ANALYSIS St™MAKT
95* Confidence Limits
Data Set
Kl 'K2
K1/K2
Intercept
s*
£
r
Steady-states
drainage tests**
1.416
1.15I to
1.680
-0.CC36 to
0.0035
7
0.992
All unsteady
drainag- tests**
1.134
1.052 to
1.217
-O.0C23 to
G.0Q23
93
0.959
Unsteady drainage
following rainfall**
1.152
1.059 to
1.246
-0.0026 5o
0.0026
79
0.955
Unsteady drainage
with presaturation**
1.049
0.927 to
1.171
-0.0035 to
0.0038
14
0.991
Estimates from
Boassines*; solutiont
0.614
0.285 to
0.942
-0.00"6 to
0.0071
6
0.960
Estimates from
HELP equationf
0.831
0.406 to
1.255
-0.010C to
0.0093
6
0.964
* N • Number of values la data see.
** Kl - Hyditulic conductivity value using BELP equation.
K2 ¦ Hydraulic conductivity value using Soussinesq solution,
t K1 * Hydraulic conductivity value for steady-state draisage.
K2 • Hydraulic conductivity value for unsteacy drainage.
The 95-percent confidence llaits for the ratic and the intercept ara also
presented in tables B-l and 8-2 to show the significance of the results.
The ratio of the HELP te iate (KS) to the Soussinesq estimate (KB) for
steady-state drainage is seae*. -ically different at 95-percent percrat con-
fidence from the ratio for unsteady drainage. The ratio for steady-state
drainage vaa 1.416 while it was 1.134 for unsteady drainage. The rrgressions
ere shown in Figure B-2. The difference in the ratios for unsteady drainage
following rainfall and fron presaturated sands as shown in Figure B-3 was net
significant at 95-percent confidence, but the ratio for presaturati^n was
smaller and not significantly different from l.OCO. The HELP hydraulic
conductivity values are significantly greater than the values estimated by the
Soussinesq solution. As presented in Section 6, this means that the lateral
drainage rate is underestimated bj the HELP model in comparison to the
Boussioesq model by j»n average of 29 percent for steady-state draisage and
12 percent for unsteady drainage.
Six par^eters w»re examined in the laboratory drair.age tests: type of
sar.d, slope, depth of saturation, draisage length, rainfall duration and
-------
032
a HELP
o BCUSSINESO
HELP
0)6-
BOUSSINESQ
CD
008-
.016
UNSTEADY K (cm/s-c)
.024
.032
.008
Figure B-l. Comparisons of hydraulic conductivity values for ste£dy-state
drainage to values for unsteady drainage as estimated by the
HEI.P equation and the numerical Bousslnesq solution.
-------
TABLE 1-2. HYDRAULIC CONDUCTIVITY REGRESSION ANALYSIS
FOR UNSTEADY DRAINAGE FOLLOWING SAINTALL


SIX Confi
dence Lir.its

2
r
Data Set
KB/ICS*
KH 'KB
Intercept
N**
Fine Sand
1.053
0.99* to
1.112
-0.0C56 to
0.0079
42
0.993
Coarse Sand
1.162
0.930 to
1.395
-0.0082 to
0.0096
37
0.953
Short Model
1.157
1.018 to
1.295
-0.0045 to
0.0036
43
0.950
Long Model
1.145
1.023 to
1.266
-0.0025 to
0.0037
36
0,964
21 Slope
1.033
1.011 to
1.055
~0.0007 to
0.0007
39
0.999
51 Slope
1.216
0.800 to
1.632
-0.0083 to
0.0112
12
0.940
102 Slope
1.427
1.235 to
1.619
-0.0063 to
0.0026
28
0.95?
Average Depth
of 0" to 7"
1.195
0.946 to
1.443
-0.0087 to
0.0113
27
0.951
Average Depth
of 8" to 14"
1.039
0.935 to
1.143
-0.0018 to
0.0024
31
0.984
Average Depth
of 15" to 20"
i.022
C.937 to
1.106
-0.0007 to
0.0013
21
0.991
y - 0" to 7"
21 Slope
1.044
0.995 to
1.092
-0.0025 to
0.0023
11
0.999
y - 0" to 7"
102 Slope
1.503
1.016 to
1.989
-0.0164 to
0.0146
13
0.964
y - 8" to 14"
2Z Slope
1.003
0.964 to
1.042
-0.0008 to
0.0010
16
0,999
y - S" to 14"
102 Slope
1.136
0.355 to
1.416
-0,0039 to
0.0039
11
0.957
y - 15" to 20"
21 Slope
1.024
0.903 to
1.145
-0.0013 to
0.0022
12
0.992
y - 15" to 20"
102 Slope
0.935
0.664 to
1.206
-0.0016 to
0.0016
e
0.999
* ra « Hydraulic conductivity value using HELP equation.
O • Hydraulic conductivity value using Bouesinesq solution.
** N - Number of "aluea in data sat.
95
-------
. 1
\£j
OS
.08-
a
a)
(/>
E
-------
. 1-1
~ FOLLOWING RAiNFALL
o PRCSATURAT10N
08"
o
Q)
(/)
E
O
Q.
LxJ
06-
FOLLOWINQ RAINFALL

PRESA TURA TION
		,	!	
.030	.045	.060
BOUSSJNESQ K (cm/sec)
Figure B-3. Comparisons of hydraulic conductivity astlmates by tlie
HELP equation to estimates by the numerical Bousslnesq
solution for unsteady drainage following rainfall and
ftoni presaturated sands.
075
-------
rainfall Intensity. Inspection of the data indicated that rainfall duration
and intensity had no effect on the drainage rate for a given sand, slope,
length and saturated depth as depicted by the hydraulic conductivity esti-
mates. This was verified by performing an unequal two-way analysis of
variance (ANOVA) using slope as the blocks and either rainfall duration cr
ittensfty as.the treatments. Blocks are the different levels of a variable
that produced variance in the measured result. The levels of slope for the
blocks vere 2, 5 and 10 percent. Treatments are the different levels of the
second variable that produce variance (6). S.trpe was selected for the blocks
siince it shewed the greatest effects on the ratios of drainage rates presented
in Section 6 and the ratios of hydraulic conductivity valt es as shown in
Table B—2 .
The effects of the remaining four paranetera on the ratio of \ie hydrau-
lic conductivity estimates were examined by two methods. The first method was
linear regression of the HHP-generated and Boussinesq-generated estimates of
hydraulic conductivity. This method does not examine interaction effects and
can yield erroneous results due to interactions and cross-correlations.
Therefore, unequal two- and three-way ANOVAs on the 79 ratios of the hydraulic
conductivity estimates for unsteady drainage following rainfall were used as
the second rethed because of their ability Co accoun. for interactions between
variables.
The results of the linear regression analyses for '.ype of sand, drainage
length, slope, average depth of saturation and average depth of saturation at
a given slope are presented in Table B-2. The ratio for estimates generated
with data from drainage tests using fine sand was 1.0S3, while it was 1.162
for estimates for coarse sand. The regressions are shown in Figure 1-4. How-
ever, the confidence interval for the coarse sand ratio was large and included
the confidence limits for the fine sand ratio so the type of sand did not sig-
nificantly affect the ratio. Eowever, since the size of the confidence inter-
vals differed by a factor of 3, the type of sand may have an interaction with
at least one of the other variables. The ratios for short model estimates and
long model estimates are very Jiailar as shown in Figure 1-3 and therefore the
drainage length does not significantly affect the estimates. The ratio of
KH/O increased significantly with increasing slope as shown in Figure B-6,
lr-dicatlng chat the HELP model underestimates the effects of slope in increas-
ing the drainage rate as predicted by the Bcussinesq solution. The effect of
depth of saturation is shown In Figure 3-7 where the hydraulic conductivity
estimates were divided into three groups based on the depth of saturation for
which the estimate was generated. These ratios were not significantly differ-
ent because the confidence Interval for tha ratio at depths of saturation
ranging from 0 to 7 inches wra such larger than the others, As with the type
of sand, an interaction may he occurring with another variable.
The interaction of slop* and depth of saturation wa® examined by perform-
ing separate regressions for the effect of depth of saturation using estimates
of hydraulic conductivity for drainage tests at 2-percent and 10-percent
slopes. Figures B-8 and B-? show the regressions for 2-percent and 10-percent
slopes, respectively. At 2-percent slope the ratioi did not differ at the
three ranges of depth of saturation. At 10-percent slope KH/KS decreased sig-
nificantly with increasing depth of saturation. Consequently, the MEL?
98
-------
O FINE SAND
a COARSE SAND
.08-
u
QJ
" . Ofi-
E
U
COARSE SAND
FINE S.tND
	I
LlJ
m
.02-
.030
.060
.015
.075
NOUSSINESQ K (cm/sec)
Figure B-A. Comparisons of hydraulic conductivity estimates by the
HELP equation to estimates by the numerical Bousslnesq
solution for unsteady drainage from fine and coarse
Bands.
-------
O SHORT MODEL
~ LONG MODEL
.00"
o o
(J
Q)
\ .06-
E
(\
SHORT MOOEL
LONG MODEL
Q_

UJ
~C
.045
(cm/
.030
BOUSSINESQ K (cm/sec)
.015
Figure B-5. Comparisons of hydraulic conductivity estimates by the HF.l.P
equation to estimates by the numerical Rousslnesq solution
for unsteady drainage from the short and long physical models.
-------
Q 27. SLOPE
o 5Z SLOPE
* m SLOPE
. 00H
/-N
u
01
\ .06-
E
a
10% SLOPE
oO
" 2'/. SLOPE
UJ
:c
.02-
o-
.060
.075
.015
.030
B0USS1NESQ K (cm/sec)
m/sec)
Figure B-6. Comparisons of hydraulic conductivity estimates by the IIEI.P equation
to estimates by the numerical Bousslnesq solution for unsteady drainage
froia models at 2-, 5- and 10-percent slopes.
-------
TO 20
r ••
.08-
u
Ol
\ .06-
E
U
TO 7
.04-
Q_
111
X
.02-
. 060
. 075
.015
. 030	.045
DOUSSINFSCI K (cm/sec)
Figure B-7. Comportuona of hyiir/vjllc conductivity estimates by the HEI.P equation
to estimates by the numerical RouBslnesq solution for unBtendy
drainage ««t depths of saturation ranging from 0 to 7 in., 8 to
14 In., and 15 to 20 In.
-------
.08
0" 10
.06-
9=0' TO 7r	^
o
OJ
U)
E
a
~ .04"
TO 14
15' TO 20
Q.
la I
.02-
.0)5
.030
DOUSSINESQ K (cm/sec)
.060
.045
(cm/se
c
Figure B-fl. Comparisons of hydraulic conductivity estimated hy the HELP equation
to estimates by" the numerical Bousslnesq solution for unoteady dralnag'.
from models at 2-pt.-cert slope wlfh depths of saturation ranging from
0 to 7 In., 8 to 14 in., and 15 to 20 In.
-------
. 1
o
i-
0
01
<7>
N
&
U
a.
U)
.00-
06-
. oH
.02-
° y
o y
A y
0" TO 7"
8" TO 14"
15" TO 20"

y-16 TO 20
	1	
.020
BOUSSINESQ K
.030
(cm/sec)
Figure B-9. Comparisons of hydraulic conductivity estimates Hy the HELP equation
to astlmatie by the numerical Bo'iselnesq solution for unsteady drainage
from modelt* at 10-percent slope with depths of saturation ranging from
0 to 7 in-, H to !4 In., and 15 to 20 In.
-------
equation underestimated lateral drainage at small depths and large slopes
compared to the Boussinesq solution but agreed well at depths above 14 in.
Additional comparisons of the effects of design parameters and their
interactions on K3/KB were made using unequal thr?e-vay ANOVA. Use ;f ANOVA
on the ?9 ratios for unsteady drainage relieving rainfall indicated that the
main effects of slope, depth of saturation and drainage length and the
interactive effects of slope with drainage length and slope with depth (listed
In their order of importance) were all significant at 95-percent confidence.
tf3iag AHOVA on the 42 ratios froo the tests on fine sand, only the effect of
slope was significant at 95-percent confidence. For the 37 values from tests
an coarse sand, the interactive effect of slope with drainage length and the
mtfin effect of slope vera significant at 95-percent confidence.
Results of ANOVAs on the data sets divided by type of sand Indicates that
interaction occurred between the type of sand and the other effects such as
death of saturation, drainage length, and slope with depth, since these vari-
ables were not significant upen dividing, the data set3. Therefore, an addi-
tional unequal three-way ANOVA was performed examining the effects of type of
sand, slope and drainage length using all 79 ratios for unsteady drainage
following rainfall. Depth of saturation was replaced by type of sand in the
analysis since depth of saturation was not significant in either of the two
ATOVAs performed on data for one type of sand. T>e results indicate that the
interactive effect of slope with drainage length and the main effects of
slope, type of sand and drainage length (listeJ in the order of importance)
are significant at 95-p«reent coofidince.
In cone.1 usion, the HELP equation underpredicts the lateral drainage rate
in comparison to the numerical solution of the Soussineaq equation by about
12 percent for unsteady drainage and by about 29 percent for steady-state
drainage. The actual underpredicticn Is a function of slope, product of the
slope and drainage length, type of sand, and drainage length. The tvo models
treat these effects in significantly different ways but, nevertheless, fre-
quently produce similar drainage rates. Under the worst conditions, the
drainage rate was unierprediced by 50 percent and overpredicted by 50 percent
during unsteady drainage. Under steady-state conditions, drainage was under-
predicted by a range of 13 to 40 pe~cent.
105
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appendix c
COMPARISONS BE7VETM LASORATORY KIASURJ^ENTS
AND NUMERICAL 3CUSSI.VESQ SOLUTIONS
The HELP lateral drainage equation was developed to approximate numerical
solutions of the one-dimensional Boussinesq equation for unsteady, unconfined,
saturated flow through porous media. Appendix B and parts of Section 6 of
this report present comparisons between che ETLP equation and the Boussinesq
solution to evaluate the validity of the EEL? equation. Therefore, it is
necessary to compare the Boussinesq solutions to the laboratory measurements.
This forss a basis for judging the significance cf the differences between the
HELP equation predictions and the laboratory aeasureaents, and between che
HELP equatien and the Boussinesq solution. This appendix presents the com-
parisons and discusses briefly how well the Boussinesq solution predicts the
drainage results. In addition, the performance of the HELP equation is com-
pared to the Boussinesq solution results to present the significance of the
differences between the HELP predictions and the laboratory aeasureaents.
Figures C-l and C-2 show the range of measured drainage rat«s aa a func-
tion of average head above eh* liner (average depth of saturation). Three
predictions by the numerical Boussinesq solution using three hydraulic conduc-
tivity conditions are shown for each test condition. The first curve is based
on the hydraulic conductivity measured in the laboratory permeameters. The
second curve was obtained by using an average of Che various hydraulic conduc-
tivity values estimated for the test condition by attempts to calibrate the
Boussio«sq solution. For the third curve, che hydraulic conductivity was
entered as a pow»r function of the average depth of saturation. This equation
vas developed by performing a curve fit of the hydraulic conductivity values
obtained by calibration. Thus# power functions are presented in Section 6.
These figures are analogous to Figures 25 and 26 in Section 6, where the
predictions were obtained using the HELP model instead of the Boussinesq
solution.
The predictions using the lasoratcry-measured hydraulic conductivity
value differ greatly from the actual results in all but one of the eight test
conditions. Predictions based on the mean hydraulic conductivity values cali-
brated for the test conditions performed better and generally showed agreement
over a narrow range of average depths. Better agreement was obtained using
che function for hydraulic conductivity, but the relationship did not fall
within the range of observations throughout the entire range of average satu-
rated depth despite all of the attempts to calibrate the Boussinesq solution.
Comparing the results shown in Figures C-l and C-2 with Figures 25 and 26
shows that tne results obtained using the HELP equation with the same
106
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<
£T
•MOAT MOOtL
•« tLOft
AVlflAGt HtAU (inches)
Pr*dict*4 RmuII*
* l«6 I
o Hot M
<
z
S
T, I." " 1
AVfRACC lt£A(l (inchus)
Fr*Jut«4 IimIIii
* lot R
Ui
~—
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oc
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f*« HOM
fT I.	!
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n
Fradidd KnuIUi
• Id* ¦
J.
u
»—
oc
X
<
a
AVJRAbf MAD (incltus*
Figure C-l. Measured drainage rate va. average head compared to numerical
Bousslneaq solutions for fine sand In both physical models
at elopes of 2 and 10 percent.
-------
tONO UOOli
am lion

UJ
»-•
•<
.15-
QC
z
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E
Pr«dicl«d flodillii
- t<4> «
o Nao) K
is ir~
AVCKACf HAD (inches)
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i
m
Ui
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Pr*dicl«d Rtiullii
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O »«- • f (y)
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10ft UOPk
.16-
Ui
*-
¦<
¦<
Aridictld RainUii
«ldiK
.01-
AVtRAW IK AO (inclias)
Figure C-2. Measured drainage rate vs. average head compared to numerical
Rousslnesq solutions for coarre sand If. both physical models at
slopes of 2 and 10 percent.
-------
hydraulic conductivity values produced results that were slightly better,
though perhaps not significantly, than the Boussinesq solution, Thti result
is surprising since the hydraulic conductivity values wete calibrated for the
Boussinesq solution and not the HELP equation. Consequently, it is apparent
that the HELP model performs as well as the Boussinesq solution. In addition,
the differences between the two methods are much smaller than the differences
between the predicted and actual results, and in some cases are smaller than
the range in the actual results.
Figures C-3 through C-6 show the measured drainage rate and the predicted
drainage rate as a function of time using the Boussinesq solution with the
power function of depth of saturation used for the hydraulic conductivity
value. Figures C-3 through C-6 also show the measured average depth of
saturation and the predicted depth as a function of time. These figures are
analogous to Figures 22 through 30 in Section 6 wher- the predicted values
were obtained using the HELP equation. The results shown in these figures
vary greatly; some show good agreement, some show poor agreement, and some
show good agreement only after infiltration cesses. The same is tru* for both
the drainage rate and the average depth of saturation. In general, t*he
results were very comparable to the results obtained using the XIL? model but
were significantly better fcr test conditions having 10-percent slope. The
HELP model yielded better results for test conditions having 2-perttnt slope.
The HELP equation predicted smaller drainage rates for a given average depth
of saturation.
Figures C-7 through C-10 show the measured and predicted head above "he
dra
-------
.11
24.
JUIM Af 0.14 IN./
9CRT NODa Af 21 SLOPE
11-
- PREOICHO
a ACTUAL
I
z
.3
i.o-m rain a? a37 m/m
sent MQOa Af IQX SLUPC
.1-
.ii-
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2
X
I
e.(H* RAIN Af a 57 IM./M)
SHORT NOOO. AT IQ1 SLOPt
— PWOICTIO
O ACTUAL
II-
• - -
14. I-M) RAIN Al 0. !4 I* /H
SHORT NOOCl AT 21 SLOP!
— WCOICTED
o ACTUAL
II-
<
TINE (Nmr«)	TIME (hart)
Figure C-3.
Measured drainage rate and average saturated depth vr. time compared to
numerical Bousslnesq solutions for tine cand In short model at aiopee of
2 and 10 percent.
-------
ft 2-m KAIN A! 0 U IK
IOC MOQII AT 101 SI.OPE
H
OBOO
%.&** ^AIN Al 0.54 1M. /Hi
IQrC ill*I Al Zt StlFi
I
UJ
i
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ioA
llMt (hour*)	FINE (haurc)
a B
it >« mi* *r a st in /»
limc Mm «i ;i aort
- W?0ICUB
a ACTUM.
To ~ la
riHf Ihotrt!
fc
106
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BM« ff&Jti A! a 54 IN. /m
LONG MOOa Af 101 SLOP!
-- PftfOfCIUi
a ACltttt.


1—Ti—R "Ttt JT~G 17
I INC &««•»>
«4
Kijjure C-4.
Measured drafnagu rate and average s.imraied deptli vs. time compared to
numerical Hhusn ltiea«| solutions fur fine Hand in J model ill .slu|iii, ;if
2 and in pcrcint.
-------
L (M« IUIN AT (L SO IN. /Ml
MM MOO. AT 21 SLOPE
.01-
o o
n-m iuin ai a» in./>«
SHOD I mXXL AI 102 SLOPE
— pf&mcTco
O ACTUAL
X
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a a a
LO-t* HA I.N AT 0. St IN. AA
SMMI MOO. AT 21 SLOPC
a-
- PREDICTED
O ACTUAL
IS-
M-
O a
T IMC (hours)
L 1-tA IUIN AT a 56 IN. AH
SHORT MlXl AI 102 9 OPE
tt-
e
&
i-
f IMC (hours)
Figure C-5. Measured drainage rate and average saturated depth va. time compared
to numerical Bousslneaq solutions for coarse sand In short model at
slopes of 2 and 10 percent.
-------
«. i-mi hasm At a ¦M in /tit
IK MGDU A! 21 &D*
0 o a o
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r SMBE ftiourc)
e z-m uin m o. 57 in. /m
lflN& MUL *1 I (LI iiQPl
PRCIICTCO
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1	U	G—S—5	£—k
1 IMC Jhour*>
C l-W RAIN A? ftSfl IM.
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pmiciro
o ACTUM.
8. M9I RAIN Al 0 17 IM. /M
iwt tmx a! iqc acre
pwnicicD
« actual
illSO
T IMC (ho,»«>
lb Jo E lu Is
IM (har*>
Menstired ilroJrate dinl average satur^t^d depth vs. t Jim: i on»|>«.roJ
ldi*l
sl
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• h ¦ 17.78* nil. PREDICTED
« K ¦ 12.63* 05*IN. ACrjM.
c fi • ;7.6r FILL. ACUA1
LH
UJ
X
k-t
z
1% SLOPE
15-
UJ
>
JO-
CD
05
<
O
^rr** °f *
DISTANCE I'RCM DRAIN (FEET)
*	S * 37.03- ORAiN. PREDICTED
•	B • 43.81* PILL PREDICTED
44. 57- ? ILL ACTUAL
uj
X
10* MUOPt
JZ
a
ns
<«
20-1
9
LU
HH
DISTANCE FROM drain (FE£T)
Figure C-7. Measured head profiles for unsteady drainage during 24-hr
rainfall test at 2-percant slope and during 6-hr rainfall
test at 10-percent slop* compared to numerical Boussinesc,
solutions for fine sand in short model.
1U
-------
40-
15-
o s • ;i50* ^?a:s. actui
c h ¦ 21,79* nu_ ACTUA^
z
5»
a
cl
-c
a
u_
?C,0	30.0
DISTANCE FROM QRAiN (FEET)
40,0
SC. 0
50.3
CFE
T
Vi-
sa
«c
a
<
• R • H.63* \UA1N. PRO ICED
¦ h • 47.56* Flu. W£CIC*£D
» h • 4
D h • 4
mo
ZC. 0	3C.0
DISTANCE. FROM DRAIN
40.3
(FEET)
53.0
60. 3
Figure C-8. Measured he*'1 profiles for unsteady drainage durinK 6-hr
rainfall test craj.artd ro numerical Bousslne?<[ solutions
for fine sand in long model &c slopes of 2 and 10 percent.
115
-------
*	h . ;9.41* DRAIN, PREDICT^]
¦ h • 10.72* f:ii. WED!CE
•	5 ¦ IKS* CHAIN, AC'Wl
£ 25-
x
JZ
z
15-
«<
QC
cz
UJ
>-
O
03
«C
ItH
o
5'
———
s'
DISTANCE FROM DRAIN (FEET)
40
%r>
Lj-J
2
z
X
z
as
a
o
m
<
o
«e
>*
Figure C-9.
•	h • 25.88* DRAIN, BREEICTiJ
F • 19. «* HLL PraiCTED
•	R • 25. 88* CfUJN. ktl-JL
a R • 19.37* PIlL ACTUAL
I0« StOrt
T
DISTANCE FRCM DRAIN
iTEET)
Measured head profiles for unsteady drainage during 6-hr rain-
fall test compared to numerical Bouaainesq solutions for
coarae sand ia short model at slopes of 2 and 10 percent.
116
-------
>
•	fi • i2. ;2" wain. f*?n:ro
•	R • 20. 88" F!lI. DIETED
•	fi • 12. 13" ORAINITIAL
Oh- 2C. Si'JiiL' ACTUAL
Cn
yj
x
LJ
Z
20-
gj
>
o
CD
«e
id-
distance FROM ZRA1N (FEET)
tn
50-
o
m
¦<
Q
C
uj
X
*	R • 4S.7T DRAIK WCTICTED
«	R • 38.9S# FILL TO3ICTE2
*	R * 45. 6fl* VtAlK ACrjAL
q	fi • m f?- Pin, actual
store
• 10.0	2G.0	3D. 0	4£LQ
DISTANCE FflOH DRAIN (FEET)
6H0
Figure C-10. Measured head profiles for unsteady drainage during 6-hr
rainfall test compared to numerical Bousalnesq solutions
for coarse sand In long model at slopes cf 2 and 10 percent*
117
-------
TECHNICAL REPORT DATA
Ifitait rtsdInsinitttont or. the ttunt bt.arr eomrtttnnt!
1. PISCO**. NO.
ZPA/GO.),'2-87/0^9
2.
3. ACCOUNT'S ACCESSION NO.
PB SI- A 17 / OH-
«. TlTi £ AND SuatiTli
Verification cf the Lateral Drainage Component o'
the HELP Model Using Physical Models
S. «6fO*T DATS
Julv 19S7
*. »0«MXC C AC AN E 2 A riON CSBi
7. *uTKO*iSI
P.R. Schroeder and R. L. Peyton
# OASAIltZAYlON NO
9 PtftFCRMINS ORC4N;2a:i0n n0. '«• kLIMINT MO.
11. CCNTAA:.T.<5fcANT f»0.
DW-96930. ".6
13. 5»onSOBinO AGENCY nam£ AND AOOAESS
Hazardous Waste Engineering Research Laboratory
Office of Research and Development
U.S Environmental Protection Agency
Lmrinnat.i . Ohio 452fi£
53. lypt Of *£Por A'.o »€AiOC COVSAtB
14. JPONSOAINO A&iNC COOI
EPA/600/12
is. sjCPlCmCsTaBy nOTIS
Project Officer; Douglas C, Ammon
16. A»ST**CT
This report describes a study conducted to verify the lateral drainage
component of the Hydrologic Evaluation of Landfill Performance (HELP) computer
model using laboratory drainage data from two la-ge-scale physical models of
landfill liner/drainage systems. Drainage tests were run to examine the effects
that drainage length, slope, hydraulic conductivity and depth of saturation have
on the lateral drainage rate. The drainage results were compared with HELP model
predictions 2nd numerical solutions of the Boussinesq equation for unsteady,
unconfirmed flow through porous media.
KIT WOK0S	eOCUMlNT ANALYSIS
J. DEISHIPT on
fe.lOINTlFKMS/OFtN (NCIS TEAMS
e. COSati Fitl-J/Grcup


•
:«. OlS''»"«WTIO* STATEMENT
Release to Public
'VmfYasYiWed
21. no. o« cages
1J 2
20. SICuhity class {T3mp*t*l
Unclassified
22. C*iCi
.
EPA 2020-1 ((*«». 4-771 «¦¦«*<«u« «oition i* «*l«k
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